# Roots and Radicals

## Simplify and Use Square Roots

### Simplify Expressions with Square Roots

Remember that when a number  is multiplied by itself, we write  and read it “n squared.” For example,  reads as “15 squared,” and 225 is called the square of 15, since .
Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because 225 is the square of 15, we can also say that 15 is a square root of 225. A number whose square is  is called a square root of .
Notice  also, so  is also a square root of 225. Therefore, both 15 and  are square roots of 225.
So, every positive number has two square roots—one positive and one negative. What if we only wanted the positive square root of a positive number? The radical sign, , denotes the positive square root. The positive square root is also called the principal square root.
We also use the radical sign for the square root of zero. Because , . Notice that zero has only one square root.
Since 15 is the positive square root of 225, we write . Fill in  to make a table of square roots you can refer to as you work this chapter.
We know that every positive number has two square roots and the radical sign indicates the positive one. We write . If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, .
When using the order of operations to simplify an expression that has square roots, we treat the radical as a grouping symbol.

### Estimate Square Roots

So far we have only considered square roots of perfect square numbers. The square roots of other numbers are not whole numbers. Look at  below.
The square roots of numbers between 4 and 9 must be between the two consecutive whole numbers 2 and 3, and they are not whole numbers. Based on the pattern in the table above, we could say that  must be between 2 and 3. Using inequality symbols, we write:

### Approximate Square Roots

There are mathematical methods to approximate square roots, but nowadays most people use a calculator to find them. Find the  key on your calculator. You will use this key to approximate square roots.
When you use your calculator to find the square root of a number that is not a perfect square, the answer that you see is not the exact square root. It is an approximation, accurate to the number of digits shown on your calculator’s display. The symbol for an approximation is  and it is read ‘approximately.’
Suppose your calculator has a 10-digit display. You would see that
If we wanted to round  to two decimal places, we would say
How do we know these values are approximations and not the exact values? Look at what happens when we square them:
Their squares are close to 5, but are not exactly equal to 5.
Using the square root key on a calculator and then rounding to two decimal places, we can find:

### Simplify Variable Expressions with Square Roots

What if we have to find a square root of an expression with a variable? Consider . Can you think of an expression whose square is ?
When we use the radical sign to take the square root of a variable expression, we should specify that  to make sure we get the principal square root.
However, in this chapter we will assume that each variable in a square-root expression represents a non-negative number and so we will not write  next to every radical.
What about square roots of higher powers of variables? Think about the Power Property of Exponents we used in Chapter 6.
If we square , the exponent will become .
How does this help us take square roots? Let’s look at a few:

### Key Concepts

1. Note that the square root of a negative number is not a real number.
2. Every positive number has two square roots, one positive and one negative. The positive square root of a positive number is the principal square root.
3. We can estimate square roots using nearby perfect squares.
4. We can approximate square roots using a calculator.
5. When we use the radical sign to take the square root of a variable expression, we should specify that  to make sure we get the principal square root.

### Practice Makes Perfect

Simplify Expressions with Square Roots
In the following exercises, simplify.
Estimate Square Roots
In the following exercises, estimate each square root between two consecutive whole numbers.
Approximate Square Roots
In the following exercises, approximate each square root and round to two decimal places.
Simplify Variable Expressions with Square Roots
In the following exercises, simplify.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?


# Roots and Radicals

## Simplify Square Roots

In the last section, we estimated the square root of a number between two consecutive whole numbers. We can say that  is between 7 and 8. This is fairly easy to do when the numbers are small enough that we can use .
But what if we want to estimate ? If we simplify the square root first, we’ll be able to estimate it easily. There are other reasons, too, to simplify square roots as you’ll see later in this chapter.
A square root is considered simplified if its radicand contains no perfect square factors.
So  is simplified. But  is not simplified, because 16 is a perfect square factor of 32.

### Use the Product Property to Simplify Square Roots

The properties we will use to simplify expressions with square roots are similar to the properties of exponents. We know that . The corresponding property of square roots says that .
We use the Product Property of Square Roots to remove all perfect square factors from a radical. We will show how to do this in .
Notice in the previous example that the simplified form of  is , which is the product of an integer and a square root. We always write the integer in front of the square root.
We could use the simplified form  to estimate . We know 5 is between 2 and 3, and  is . So  is between 20 and 30.
The next example is much like the previous examples, but with variables.
We follow the same procedure when there is a coefficient in the radical, too.
In the next example both the constant and the variable have perfect square factors.
We have seen how to use the Order of Operations to simplify some expressions with radicals. To simplify  we must simplify each square root separately first, then add to get the sum of 17.
The expression  cannot be simplified—to begin we’d need to simplify each square root, but neither 17 nor 7 contains a perfect square factor.
In the next example, we have the sum of an integer and a square root. We simplify the square root but cannot add the resulting expression to the integer.
The next example includes a fraction with a radical in the numerator. Remember that in order to simplify a fraction you need a common factor in the numerator and denominator.

### Use the Quotient Property to Simplify Square Roots

Whenever you have to simplify a square root, the first step you should take is to determine whether the radicand is a perfect square. A perfect square fraction is a fraction in which both the numerator and the denominator are perfect squares.
If the numerator and denominator have any common factors, remove them. You may find a perfect square fraction!
In the last example, our first step was to simplify the fraction under the radical by removing common factors. In the next example we will use the Quotient Property to simplify under the radical. We divide the like bases by subtracting their exponents, .
Remember the Quotient to a Power Property? It said we could raise a fraction to a power by raising the numerator and denominator to the power separately.
We can use a similar property to simplify a square root of a fraction. After removing all common factors from the numerator and denominator, if the fraction is not a perfect square we simplify the numerator and denominator separately.
Be sure to simplify the fraction in the radicand first, if possible.

### Key Concepts

1. Simplified Square Root  is considered simplified if  has no perfect-square factors.
2. Product Property of Square Roots If a, b are non-negative real numbers, then

3. Simplify a Square Root Using the Product Property To simplify a square root using the Product Property:

4. Quotient Property of Square Roots If a, b are non-negative real numbers and , then

5. Simplify a Square Root Using the Quotient Property To simplify a square root using the Quotient Property:

### Practice Makes Perfect

Use the Product Property to Simplify Square Roots
In the following exercises, simplify.
Use the Quotient Property to Simplify Square Roots
In the following exercises, simplify.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?


# Roots and Radicals

## Add and Subtract Square Roots

We know that we must follow the order of operations to simplify expressions with square roots. The radical is a grouping symbol, so we work inside the radical first. We simplify  in this way:

So if we have to add , we must not combine them into one radical.
Trying to add square roots with different radicands is like trying to add unlike terms.

Adding square roots with the same radicand is just like adding like terms. We call square roots with the same radicand like square roots to remind us they work the same as like terms.
We add and subtract like square roots in the same way we add and subtract like terms. We know that  is . Similarly we add  and the result is

### Add and Subtract Like Square Roots

Think about adding like terms with variables as you do the next few examples. When you have like radicands, you just add or subtract the coefficients. When the radicands are not like, you cannot combine the terms.
When radicals contain more than one variable, as long as all the variables and their exponents are identical, the radicals are like.

### Add and Subtract Square Roots that Need Simplification

Remember that we always simplify square roots by removing the largest perfect-square factor. Sometimes when we have to add or subtract square roots that do not appear to have like radicals, we find like radicals after simplifying the square roots.
Just like we use the Associative Property of Multiplication to simplify  and get , we can simplify  and get . We will use the Associative Property to do this in the next example.
In the next example, we will remove constant and variable factors from the square roots.

### Key Concepts

1. To add or subtract like square roots, add or subtract the coefficients and keep the like square root.
2. Sometimes when we have to add or subtract square roots that do not appear to have like radicals, we find like radicals after simplifying the square roots.

### Practice Makes Perfect

Add and Subtract Like Square Roots
In the following exercises, simplify.
Add and Subtract Square Roots that Need Simplification
In the following exercises, simplify.
Mixed Practice

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?


# Roots and Radicals

## Multiply Square Roots

### Multiply Square Roots

We have used the Product Property of Square Roots to simplify square roots by removing the perfect square factors. The Product Property of Square Roots says
We can use the Product Property of Square Roots ‘in reverse’ to multiply square roots.
Remember, we assume all variables are greater than or equal to zero.
We will rewrite the Product Property of Square Roots so we see both ways together.
So we can multiply  in this way:
Sometimes the product gives us a perfect square:
Even when the product is not a perfect square, we must look for perfect-square factors and simplify the radical whenever possible.
Multiplying radicals with coefficients is much like multiplying variables with coefficients. To multiply  we multiply the coefficients together and then the variables. The result is . Keep this in mind as you do these examples.
When we have to multiply square roots, we first find the product and then remove any perfect square factors.
The results of the previous example lead us to this property.
By realizing that squaring and taking a square root are ‘opposite’ operations, we can simplify  and get 2 right away. When we multiply the two like square roots in part (a) of the next example, it is the same as squaring.

### Use Polynomial Multiplication to Multiply Square Roots

In the next few examples, we will use the Distributive Property to multiply expressions with square roots.
We will first distribute and then simplify the square roots when possible.
When we worked with polynomials, we multiplied binomials by binomials. Remember, this gave us four products before we combined any like terms. To be sure to get all four products, we organized our work—usually by the FOIL method.
Note that some special products made our work easier when we multiplied binomials earlier. This is true when we multiply square roots, too. The special product formulas we used are shown below.
We will use the special product formulas in the next few examples. We will start with the Binomial Squares formula.
In the next two examples, we will find the product of conjugates.

### Key Concepts

1. Product Property of Square Roots If a, b are nonnegative real numbers, then

2. Special formulas for multiplying binomials and conjugates:

3. The FOIL method can be used to multiply binomials containing radicals.

### Practice Makes Perfect

Multiply Square Roots
In the following exercises, simplify.
Use Polynomial Multiplication to Multiply Square Roots
In the following exercises, simplify.
Mixed Practice
In the following exercises, simplify.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?


# Roots and Radicals

## Divide Square Roots

### Divide Square Roots

We know that we simplify fractions by removing factors common to the numerator and the denominator. When we have a fraction with a square root in the numerator, we first simplify the square root. Then we can look for common factors.
We have used the Quotient Property of Square Roots to simplify square roots of fractions. The Quotient Property of Square Roots says
Sometimes we will need to use the Quotient Property of Square Roots ‘in reverse’ to simplify a fraction with square roots.
We will rewrite the Quotient Property of Square Roots so we see both ways together. Remember: we assume all variables are greater than or equal to zero so that their square roots are real numbers.
We will use the Quotient Property of Square Roots ‘in reverse’ when the fraction we start with is the quotient of two square roots, and neither radicand is a perfect square. When we write the fraction in a single square root, we may find common factors in the numerator and denominator.
We will use the Quotient Property for Exponents, , when we have variables with exponents in the radicands.

### Rationalize a One Term Denominator

Before the calculator became a tool of everyday life, tables of square roots were used to find approximate values of square roots.  shows a portion of a table of squares and square roots. Square roots are approximated to five decimal places in this table.
If someone needed to approximate a fraction with a square root in the denominator, it meant doing long division with a five decimal-place divisor. This was a very cumbersome process.
For this reason, a process called rationalizing the denominator was developed. A fraction with a radical in the denominator is converted to an equivalent fraction whose denominator is an integer. This process is still used today and is useful in other areas of mathematics, too.
Square roots of numbers that are not perfect squares are irrational numbers. When we rationalize the denominator, we write an equivalent fraction with a rational number in the denominator.
Let’s look at a numerical example.

But we can find a fraction equivalent to  by multiplying the numerator and denominator by .
Now if we need an approximate value, we divide . This is much easier.
Even though we have calculators available nearly everywhere, a fraction with a radical in the denominator still must be rationalized. It is not considered simplified if the denominator contains a square root.
Similarly, a square root is not considered simplified if the radicand contains a fraction.
To rationalize a denominator, we use the property that . If we square an irrational square root, we get a rational number.
We will use this property to rationalize the denominator in the next example.
Always simplify the radical in the denominator first, before you rationalize it. This way the numbers stay smaller and easier to work with.

### Rationalize a Two-Term Denominator

When the denominator of a fraction is a sum or difference with square roots, we use the Product of Conjugates pattern to rationalize the denominator.
When we multiply a binomial that includes a square root by its conjugate, the product has no square roots.

### Key Concepts

1. Quotient Property of Square Roots

2. Simplified Square RootsA square root is considered simplified if there are

### Practice Makes Perfect

Divide Square Roots
In the following exercises, simplify.
Rationalize a One-Term Denominator
In the following exercises, simplify and rationalize the denominator.
Rationalize a Two-Term Denominator
In the following exercises, simplify by rationalizing the denominator.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?


# Roots and Radicals

## Solve Equations with Square Roots

### Solve Radical Equations

In this section we will solve equations that have the variable in the radicand of a square root. Equations of this type are called radical equations.
As usual, in solving these equations, what we do to one side of an equation we must do to the other side as well. Since squaring a quantity and taking a square root are ‘opposite’ operations, we will square both sides in order to remove the radical sign and solve for the variable inside.
But remember that when we write  we mean the principal square root. So  always. When we solve radical equations by squaring both sides we may get an algebraic solution that would make  negative. This algebraic solution would not be a solution to the original radical equation; it is an extraneous solution. We saw extraneous solutions when we solved rational equations, too.
Now we will see how to solve a radical equation. Our strategy is based on the relation between taking a square root and squaring.
When we use a radical sign, we mean the principal or positive root. If an equation has a square root equal to a negative number, that equation will have no solution.
If one side of the equation is a binomial, we use the binomial squares formula when we square it.
Don’t forget the middle term!
When there is a coefficient in front of the radical, we must square it, too.
Sometimes after squaring both sides of an equation, we still have a variable inside a radical. When that happens, we repeat Step 1 and Step 2 of our procedure. We isolate the radical and square both sides of the equation again.

### Use Square Roots in Applications

As you progress through your college courses, you’ll encounter formulas that include square roots in many disciplines. We have already used formulas to solve geometry applications.
We will use our Problem Solving Strategy for Geometry Applications, with slight modifications, to give us a plan for solving applications with formulas from any discipline.
We used the formula  to find the area of a rectangle with length L and width W. A square is a rectangle in which the length and width are equal. If we let s be the length of a side of a square, the area of the square is .
The formula  gives us the area of a square if we know the length of a side. What if we want to find the length of a side for a given area? Then we need to solve the equation for s.

We can use the formula  to find the length of a side of a square for a given area.
We will show an example of this in the next example.
Another application of square roots has to do with gravity.
For example, if an object is dropped from a height of 64 feet, we can find the time it takes to reach the ground by substituting  into the formula.

It would take 2 seconds for an object dropped from a height of 64 feet to reach the ground.
Police officers investigating car accidents measure the length of the skid marks on the pavement. Then they use square roots to determine the speed, in miles per hour, a car was going before applying the brakes.

### Key Concepts

1. To Solve a Radical Equation:

2. Solving Applications with Formulas

3. Area of a Square

4. Falling Objects

5. Skid Marks and Speed of a Car

### Practice Makes Perfect

Solve Radical Equations
In the following exercises, check whether the given values are solutions.
In the following exercises, solve.
Use Square Roots in Applications
In the following exercises, solve. Round approximations to one decimal place.

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?


# Roots and Radicals

## Higher Roots

### Simplify Expressions with Higher Roots

Up to now, in this chapter we have worked with squares and square roots. We will now extend our work to include higher powers and higher roots.
Let’s review some vocabulary first.
The terms ‘squared’ and ‘cubed’ come from the formulas for area of a square and volume of a cube.
It will be helpful to have a table of the powers of the integers from . See .
Notice the signs in . All powers of positive numbers are positive, of course. But when we have a negative number, the even powers are positive and the odd powers are negative. We’ll copy the row with the powers of  below to help you see this.
Earlier in this chapter we defined the square root of a number.
And we have used the notation  to denote the principal square root. So  always.
We will now extend the definition to higher roots.
We do not write the index for a square root. Just like we use the word ‘cubed’ for , we use the term ‘cube root’ for .
We refer to  to help us find higher roots.
Could we have an even root of a negative number? No. We know that the square root of a negative number is not a real number. The same is true for any even root. Even roots of negative numbers are not real numbers. Odd roots of negative numbers are real numbers.
When we worked with square roots that had variables in the radicand, we restricted the variables to non-negative values. Now we will remove this restriction.
The odd root of a number can be either positive or negative. We have seen that .
But the even root of a non-negative number is always non-negative, because we take the principal .
Suppose we start with .
How can we make sure the fourth root of −5 raised to the fourth power,  is 5? We will see in the following property.

### Use the Product Property to Simplify Expressions with Higher Roots

We will simplify expressions with higher roots in much the same way as we simplified expressions with square roots. An nth root is considered simplified if it has no factors of .
We will generalize the Product Property of Square Roots to include any integer root .
Don’t forget to use the absolute value signs when taking an even root of an expression with a variable in the radical.

### Use the Quotient Property to Simplify Expressions with Higher Roots

We can simplify higher roots with quotients in the same way we simplified square roots. First we simplify any fractions inside the radical.
Previously, we used the Quotient Property ‘in reverse’ to simplify square roots. Now we will generalize the formula to include higher roots.
If the fraction inside the radical cannot be simplified, we use the first form of the Quotient Property to rewrite the expression as the quotient of two radicals.

### Add and Subtract Higher Roots

We can add and subtract higher roots like we added and subtracted square roots. First we provide a formal definition of like radicals.
Like radicals have the same index and the same radicand.
1.  and  are like radicals.
2.  and  are not like radicals. The radicands are different.
3.  and  are not like radicals. The indices are different.
We add and subtract like radicals in the same way we add and subtract like terms. We can add  and the result is .
When an expression does not appear to have like radicals, we will simplify each radical first. Sometimes this leads to an expression with like radicals.

### Key Concepts

1. Properties of
2.  when  is an even number and

3.  is considered simplified if a has no factors of .
4. Product Property of

5. Quotient Property of

6. To combine like radicals, simply add or subtract the coefficients while keeping the radical the same.

### Practice Makes Perfect

Simplify Expressions with Higher Roots
In the following exercises, simplify.
Use the Product Property to Simplify Expressions with Higher Roots
In the following exercises, simplify.
Use the Quotient Property to Simplify Expressions with Higher Roots
In the following exercises, simplify.
Add and Subtract Higher Roots
In the following exercises, simplify.
Mixed Practice
In the following exercises, simplify.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?


# Roots and Radicals

## Rational Exponents

### Simplify Expressions with

Rational exponents are another way of writing expressions with radicals. When we use rational exponents, we can apply the properties of exponents to simplify expressions.
The Power Property for Exponents says that  when m and n are whole numbers. Let’s assume we are now not limited to whole numbers.
Suppose we want to find a number p such that . We will use the Power Property of Exponents to find the value of p.

But we know also . Then it must be that .
This same logic can be used for any positive integer exponent n to show that .
There will be times when working with expressions will be easier if you use rational exponents and times when it will be easier if you use radicals. In the first few examples, you’ll practice converting expressions between these two notations.
In the next example, you may find it easier to simplify the expressions if you rewrite them as radicals first.
Be careful of the placement of the negative signs in the next example. We will need to use the property  in one case.

### Simplify Expressions with

Let’s work with the Power Property for Exponents some more.
Suppose we raise  to the power m.

Now suppose we take  to the  power.

Which form do we use to simplify an expression? We usually take the root first—that way we keep the numbers in the radicand smaller.
Remember that . The negative sign in the exponent does not change the sign of the expression.

### Use the Laws of Exponents to Simplify Expressions with Rational Exponents

The same laws of exponents that we already used apply to rational exponents, too. We will list the Exponent Properties here to have them for reference as we simplify expressions.
When we multiply the same base, we add the exponents.
We will use the Power Property in the next example.
The Quotient Property tells us that when we divide with the same base, we subtract the exponents.
Sometimes we need to use more than one property. In the next two examples, we will use both the Product to a Power Property and then the Power Property.
We will use both the Product and Quotient Properties in the next example.

### Key Concepts

1. Summary of Exponent Properties
2. If  are real numbers and  are rational numbers, then

### Section Exercises

### Practice Makes Perfect

Simplify Expressions with
In the following exercises, write as a radical expression.
In the following exercises, write with a rational exponent.
In the following exercises, simplify.
Simplify Expressions with
In the following exercises, write with a rational exponent.
In the following exercises, simplify.
Use the Laws of Exponents to Simplify Expressions with Rational Exponents
In the following exercises, simplify.

### Everyday Math

### Writing Exercises

### Chapter 9 Review Exercises

### Simplify and Use Square Roots

Simplify Expressions with Square Roots
In the following exercises, simplify.
Estimate Square Roots
In the following exercises, estimate each square root between two consecutive whole numbers.
Approximate Square Roots
In the following exercises, approximate each square root and round to two decimal places.
Simplify Variable Expressions with Square Roots
In the following exercises, simplify.

### Simplify Square Roots

Use the Product Property to Simplify Square Roots
In the following exercises, simplify.
Use the Quotient Property to Simplify Square Roots
In the following exercises, simplify.

### Add and Subtract Square Roots

Add and Subtract Like Square Roots
In the following exercises, simplify.
Add and Subtract Square Roots that Need Simplification
In the following exercises, simplify.

### Multiply Square Roots

Multiply Square Roots
In the following exercises, simplify.
Use Polynomial Multiplication to Multiply Square Roots
In the following exercises, simplify.

### Divide Square Roots

Divide Square Roots
In the following exercises, simplify.
Rationalize a One Term Denominator
In the following exercises, rationalize the denominator.
Rationalize a Two Term Denominator
In the following exercises, rationalize the denominator.

### Solve Equations with Square Roots

Solve Radical Equations
In the following exercises, solve the equation.
Use Square Roots in Applications
In the following exercises, solve. Round approximations to one decimal place.

### Higher Roots

Simplify Expressions with Higher Roots
In the following exercises, simplify.
Use the Product Property to Simplify Expressions with Higher Roots
In the following exercises, simplify.
Use the Quotient Property to Simplify Expressions with Higher Roots
In the following exercises, simplify.
Add and Subtract Higher Roots
In the following exercises, simplify.

### Rational Exponents

Simplify Expressions with
In the following exercises, write as a radical expression.
In the following exercises, write with a rational exponent.
In the following exercises, simplify.
Simplify Expressions with
In the following exercises, write with a rational exponent.
In the following exercises, simplify.
Use the Laws of Exponents to Simplify Expressions with Rational Exponents
In the following exercises, simplify.

### Practice Test

In the following exercises, simplify.
In the following exercises, rationalize the denominator.
In the following exercises, solve.
In the following exercise, solve.


# Quadratic Equations

## Introduction

The trajectories of fireworks are modeled by quadratic equations. The equations can be used to predict the maximum height of a firework and the number of seconds it will take from launch to explosion. In this chapter, we will study the properties of quadratic equations, solve them, graph them, and see how they are applied as models of various situations.


# Quadratic Equations

## Solve Quadratic Equations Using the Square Root Property

Quadratic equations are equations of the form , where . They differ from linear equations by including a term with the variable raised to the second power. We use different methods to solve quadratic equations than linear equations, because just adding, subtracting, multiplying, and dividing terms will not isolate the variable.
We have seen that some quadratic equations can be solved by factoring. In this chapter, we will use three other methods to solve quadratic equations.

### Solve Quadratic Equations of the Form ax2 = k Using the Square Root Property

We have already solved some quadratic equations by factoring. Let’s review how we used factoring to solve the quadratic equation .
We can easily use factoring to find the solutions of similar equations, like  and , because 16 and 25 are perfect squares. But what happens when we have an equation like ? Since 7 is not a perfect square, we cannot solve the equation by factoring.
These equations are all of the form .We defined the square root of a number in this way:
This leads to the Square Root Property.
Notice that the Square Root Property gives two solutions to an equation of the form : the principal square root of  and its opposite. We could also write the solution as .
Now, we will solve the equation  again, this time using the Square Root Property.
What happens when the constant is not a perfect square? Let’s use the Square Root Property to solve the equation .
To use the Square Root Property, the coefficient of the variable term must equal 1. In the next example, we must divide both sides of the equation by 5 before using the Square Root Property.
The Square Root Property started by stating, ‘If , and ’. What will happen if ? This will be the case in the next example.
Remember, we first isolate the quadratic term and then make the coefficient equal to one.
The solutions to some equations may have fractions inside the radicals. When this happens, we must rationalize the denominator.

### Solve Quadratic Equations of the Form a(x − h)2 = k Using the Square Root Property

We can use the Square Root Property to solve an equation like , too. We will treat the whole binomial, , as the quadratic term.
Remember, when we take the square root of a fraction, we can take the square root of the numerator and denominator separately.
We will start the solution to the next example by isolating the binomial.
The left sides of the equations in the next two examples do not seem to be of the form . But they are perfect square trinomials, so we will factor to put them in the form we need.

### Key Concepts

1. Square Root PropertyIf , and , then .

### Practice Makes Perfect

Solve Quadratic Equations of the form  Using the Square Root Property
In the following exercises, solve the following quadratic equations.
Solve Quadratic Equations of the Form  Using the Square Root Property
In the following exercises, solve the following quadratic equations.
Mixed Practice
In the following exercises, solve using the Square Root Property.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ If most of your checks were:
…confidently: Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help: This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no-I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.


# Quadratic Equations

## Solve Quadratic Equations by Completing the Square

So far, we have solved quadratic equations by factoring and using the Square Root Property. In this section, we will solve quadratic equations by a process called ‘completing the square.’

### Complete The Square of a Binomial Expression

In the last section, we were able to use the Square Root Property to solve the equation  because the left side was a perfect square.
We also solved an equation in which the left side was a perfect square trinomial, but we had to rewrite it the form  in order to use the square root property.
What happens if the variable is not part of a perfect square? Can we use algebra to make a perfect square?
Let’s study the binomial square pattern we have used many times. We will look at two examples.
We can use this pattern to “make” a perfect square.
We will start with the expression . Since there is a plus sign between the two terms, we will use the  pattern.
Notice that the first term of  is a square, .
We now know .
What number can we add to  to make a perfect square trinomial?
The middle term of the Binomial Squares Pattern, , is twice the product of the two terms of the binomial. This means twice the product of  and some number is . So, two times some number must be six. The number we need is  The second term in the binomial,  must be 3.
We now know .
Now, we just square the second term of the binomial to get the last term of the perfect square trinomial, so we square three to get the last term, nine.
We can now factor to
So, we found that adding nine to  ‘completes the square,’ and we write it as .

### Solve Quadratic Equations of the Form x2 + bx + c = 0 by completing the square

In solving equations, we must always do the same thing to both sides of the equation. This is true, of course, when we solve a quadratic equation by completing the square, too. When we add a term to one side of the equation to make a perfect square trinomial, we must also add the same term to the other side of the equation.
For example, if we start with the equation  and we want to complete the square on the left, we will add nine to both sides of the equation.
Then, we factor on the left and simplify on the right.
Now the equation is in the form to solve using the Square Root Property. Completing the square is a way to transform an equation into the form we need to be able to use the Square Root Property.
In the previous example, there was no real solution because  was equal to a negative number.
We will start the next example by isolating the variable terms on the left side of the equation.
To solve the next equation, we must first collect all the variable terms to the left side of the equation. Then, we proceed as we did in the previous examples.
Notice that the left side of the next equation is in factored form. But the right side is not zero, so we cannot use the Zero Product Property. Instead, we multiply the factors and then put the equation into the standard form to solve by completing the square.

### Solve Quadratic Equations of the form ax2 + bx + c = 0 by completing the square

The process of completing the square works best when the leading coefficient is one, so the left side of the equation is of the form . If the  term has a coefficient, we take some preliminary steps to make the coefficient equal to one.
Sometimes the coefficient can be factored from all three terms of the trinomial. This will be our strategy in the next example.
To complete the square, the leading coefficient must be one. When the leading coefficient is not a factor of all the terms, we will divide both sides of the equation by the leading coefficient. This will give us a fraction for the second coefficient. We have already seen how to complete the square with fractions in this section.

### Key Concepts

1. Binomial Squares Pattern If  are real numbers,

2. Complete a SquareTo complete the square of :

### Practice Makes Perfect

Complete the Square of a Binomial Expression
In the following exercises, complete the square to make a perfect square trinomial. Then, write the result as a binomial squared.
Solve Quadratic Equations of the Form  by Completing the Square
In the following exercises, solve by completing the square.
Solve Quadratic Equations of the Form  by Completing the Square
In the following exercises, solve by completing the square.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?


# Quadratic Equations

## Solve Quadratic Equations Using the Quadratic Formula

When we solved quadratic equations in the last section by completing the square, we took the same steps every time. By the end of the exercise set, you may have been wondering ‘isn’t there an easier way to do this?’ The answer is ‘yes.’ In this section, we will derive and use a formula to find the solution of a quadratic equation.
We have already seen how to solve a formula for a specific variable ‘in general’ so that we would do the algebraic steps only once and then use the new formula to find the value of the specific variable. Now, we will go through the steps of completing the square in general to solve a quadratic equation for x. It may be helpful to look at one of the examples at the end of the last section where we solved an equation of the form  as you read through the algebraic steps below, so you see them with numbers as well as ‘in general.’

This last equation is the Quadratic Formula.
To use the Quadratic Formula, we substitute the values of  into the expression on the right side of the formula. Then, we do all the math to simplify the expression. The result gives the solution(s) to the quadratic equation.
If you say the formula as you write it in each problem, you’ll have it memorized in no time. And remember, the Quadratic Formula is an equation. Be sure you start with ‘’.
When we solved quadratic equations by using the Square Root Property, we sometimes got answers that had radicals. That can happen, too, when using the Quadratic Formula. If we get a radical as a solution, the final answer must have the radical in its simplified form.
We cannot take the square root of a negative number. So, when we substitute , , and  into the Quadratic Formula, if the quantity inside the radical is negative, the quadratic equation has no real solution. We will see this in the next example.
The quadratic equations we have solved so far in this section were all written in standard form, . Sometimes, we will need to do some algebra to get the equation into standard form before we can use the Quadratic Formula.
When we solved linear equations, if an equation had too many fractions we ‘cleared the fractions’ by multiplying both sides of the equation by the LCD. This gave us an equivalent equation—without fractions—to solve. We can use the same strategy with quadratic equations.
Think about the equation . We know from the Zero Products Principle that this equation has only one solution: .
We will see in the next example how using the Quadratic Formula to solve an equation with a perfect square also gives just one solution.

### Use the Discriminant to Predict the Number of Solutions of a Quadratic Equation

When we solved the quadratic equations in the previous examples, sometimes we got two solutions, sometimes one solution, sometimes no real solutions. Is there a way to predict the number of solutions to a quadratic equation without actually solving the equation?
Yes, the quantity inside the radical of the Quadratic Formula makes it easy for us to determine the number of solutions. This quantity is called the discriminant.
Let’s look at the discriminant of the equations in , , and , and the number of solutions to those quadratic equations.
When the discriminant is positive  the quadratic equation has two solutions.
When the discriminant is zero  the quadratic equation has one solution.
When the discriminant is negative  the quadratic equation has no real solutions.

### Identify the Most Appropriate Method to Use to Solve a Quadratic Equation

We have used four methods to solve quadratic equations:
1. Factoring
2. Square Root Property
3. Completing the Square
4. Quadratic Formula
You can solve any quadratic equation by using the Quadratic Formula, but that is not always the easiest method to use.
What about the method of completing the square? Most people find that method cumbersome and prefer not to use it. We needed to include it in this chapter because we completed the square in general to derive the Quadratic Formula. You will also use the process of completing the square in other areas of algebra.

### Key Concepts

1. Quadratic Formula The solutions to a quadratic equation of the form   are given by the formula:

2. Solve a Quadratic Equation Using the Quadratic Formula
To solve a quadratic equation using the Quadratic Formula.

3. Using the Discriminant,
For a quadratic equation of the form

4. To identify the most appropriate method to solve a quadratic equation:

### Practice Makes Perfect

Solve Quadratic Equations Using the Quadratic Formula
In the following exercises, solve by using the Quadratic Formula.
Use the Discriminant to Predict the Number of Solutions of a Quadratic Equation
In the following exercises, determine the number of solutions to each quadratic equation.
Identify the Most Appropriate Method to Use to Solve a Quadratic Equation
In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?


# Quadratic Equations

## Solve Applications Modeled by Quadratic Equations

### Solve Applications of the Quadratic Formula

We solved some applications that are modeled by quadratic equations earlier, when the only method we had to solve them was factoring. Now that we have more methods to solve quadratic equations, we will take another look at applications. To get us started, we will copy our usual Problem Solving Strategy here so we can follow the steps.
We have solved number applications that involved consecutive even integers and consecutive odd integers by modeling the situation with linear equations. Remember, we noticed each even integer is 2 more than the number preceding it. If we call the first one n, then the next one is . The next one would be  or . This is also true when we use odd integers. One set of even integers and one set of odd integers are shown below.
Some applications of consecutive odd integers or consecutive even integers are modeled by quadratic equations. The notation above will be helpful as you name the variables.
We will use the formula for the area of a triangle to solve the next example.
Recall that, when we solve geometry applications, it is helpful to draw the figure.
In the two preceding examples, the number in the radical in the Quadratic Formula was a perfect square and so the solutions were rational numbers. If we get an irrational number as a solution to an application problem, we will use a calculator to get an approximate value.
The Pythagorean Theorem gives the relation between the legs and hypotenuse of a right triangle. We will use the Pythagorean Theorem to solve the next example.
The height of a projectile shot upwards is modeled by a quadratic equation. The initial velocity, , propels the object up until gravity causes the object to fall back down.
We can use the formula for projectile motion to find how many seconds it will take for a firework to reach a specific height.

### Key Concepts

1. Area of a Triangle For a triangle with base, , and height, , the area, , is given by the formula:

2. Pythagorean Theorem In any right triangle, where  and  are the lengths of the legs, and  is the length of the hypothenuse,

3. Projectile motion The height in feet, , of an object shot upwards into the air with initial velocity, , after  seconds can be modeled by the formula:

### Practice Makes Perfect

Solve Applications of the Quadratic Formula
In the following exercises, solve by using methods of factoring, the square root principle, or the Quadratic Formula. Round your answers to the nearest tenth.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?


# Quadratic Equations

## Graphing Quadratic Equations in Two Variables

### Recognize the Graph of a Quadratic Equation in Two Variables

We have graphed equations of the form . We called equations like this linear equations because their graphs are straight lines.
Now, we will graph equations of the form . We call this kind of equation a quadratic equation in two variables.
Just like we started graphing linear equations by plotting points, we will do the same for quadratic equations.
Let’s look first at graphing the quadratic equation . We will choose integer values of  between  and 2 and find their  values. See .
Notice when we let  and , we got the same value for .
The same thing happened when we let  and .
Now, we will plot the points to show the graph of . See .
The graph is not a line. This figure is called a parabola. Every quadratic equation has a graph that looks like this.
In  you will practice graphing a parabola by plotting a few points.
How do the equations  and  differ? What is the difference between their graphs? How are their graphs the same?
All parabolas of the form  open upwards or downwards. See .
Notice that the only difference in the two equations is the negative sign before the  in the equation of the second graph in . When the  term is positive, the parabola opens upward, and when the  term is negative, the parabola opens downward.

### Find the Axis of Symmetry and Vertex of a Parabola

Look again at . Do you see that we could fold each parabola in half and that one side would lie on top of the other? The ‘fold line’ is a line of symmetry. We call it the axis of symmetry of the parabola.
We show the same two graphs again with the axis of symmetry in blue. See .
The equation of the axis of symmetry can be derived by using the Quadratic Formula. We will omit the derivation here and proceed directly to using the result. The equation of the axis of symmetry of the graph of  is
So, to find the equation of symmetry of each of the parabolas we graphed above, we will substitute into the formula .
Look back at . Are these the equations of the dashed red lines?
The point on the parabola that is on the axis of symmetry is the lowest or highest point on the parabola, depending on whether the parabola opens upwards or downwards. This point is called the vertex of the parabola.
We can easily find the coordinates of the vertex, because we know it is on the axis of symmetry. This means its x-coordinate is . To find the y-coordinate of the vertex, we substitute the value of the x-coordinate into the quadratic equation.

### Find the Intercepts of a Parabola

When we graphed linear equations, we often used the x- and y-intercepts to help us graph the lines. Finding the coordinates of the intercepts will help us to graph parabolas, too.
Remember, at the  the value of  is zero. So, to find the y-intercept, we substitute  into the equation.
Let’s find the y-intercepts of the two parabolas shown in the figure below.
At an , the value of  is zero. To find an x-intercept, we substitute  into the equation. In other words, we will need to solve the equation  for .
But solving quadratic equations like this is exactly what we have done earlier in this chapter.
We can now find the x-intercepts of the two parabolas shown in .
First, we will find the  with equation .
Now, we will find the x-intercepts of the parabola with equation .
We will use the decimal approximations of the x-intercepts, so that we can locate these points on the graph.
Do these results agree with our graphs? See .
In this chapter, we have been solving quadratic equations of the form . We solved for  and the results were the solutions to the equation.
We are now looking at quadratic equations in two variables of the form . The graphs of these equations are parabolas. The x-intercepts of the parabolas occur where .
For example:
The solutions of the quadratic equation are the  values of the x-intercepts.
Earlier, we saw that quadratic equations have 2, 1, or 0 solutions. The graphs below show examples of parabolas for these three cases. Since the solutions of the equations give the x-intercepts of the graphs, the number of x-intercepts is the same as the number of solutions.
Previously, we used the discriminant to determine the number of solutions of a quadratic equation of the form . Now, we can use the discriminant to tell us how many x-intercepts there are on the graph.
Before you start solving the quadratic equation to find the values of the x-intercepts, you may want to evaluate the discriminant so you know how many solutions to expect.

### Graph Quadratic Equations in Two Variables

Now, we have all the pieces we need in order to graph a quadratic equation in two variables. We just need to put them together. In the next example, we will see how to do this.
We were able to find the x-intercepts in the last example by factoring. We find the x-intercepts in the next example by factoring, too.
For the graph of , the vertex and the x-intercept were the same point. Remember how the discriminant determines the number of solutions of a quadratic equation? The discriminant of the equation  is 0, so there is only one solution. That means there is only one x-intercept, and it is the vertex of the parabola.
How many x-intercepts would you expect to see on the graph of ?
Finding the y-intercept by substituting  into the equation is easy, isn’t it? But we needed to use the Quadratic Formula to find the x-intercepts in . We will use the Quadratic Formula again in the next example.

### Solve Maximum and Minimum Applications

Knowing that the vertex of a parabola is the lowest or highest point of the parabola gives us an easy way to determine the minimum or maximum value of a quadratic equation. The y-coordinate of the vertex is the minimum y-value of a parabola that opens upward. It is the maximum y-value of a parabola that opens downward. See .
We have used the formula
to calculate the height in feet, , of an object shot upwards into the air with initial velocity, , after  seconds.
This formula is a quadratic equation in the variable , so its graph is a parabola. By solving for the coordinates of the vertex, we can find how long it will take the object to reach its maximum height. Then, we can calculate the maximum height.

### Key Concepts

1. The graph of every quadratic equation is a parabola.
2. Parabola Orientation For the quadratic equation , if

3. Axis of Symmetry and Vertex of a Parabola For a parabola with equation :

4. Find the Intercepts of a Parabola To find the intercepts of a parabola with equation :
5. To Graph a Quadratic Equation in Two Variables

6. Minimum or Maximum Values of a Quadratic Equation

### Section Exercises

### Practice Makes Perfect

Recognize the Graph of a Quadratic Equation in Two Variables
In the following exercises, graph:
In the following exercises, determine if the parabola opens up or down.
Find the Axis of Symmetry and Vertex of a Parabola
In the following exercises, find ⓐ the axis of symmetry and ⓑ the vertex.
Find the Intercepts of a Parabola
In the following exercises, find the x- and y-intercepts.
Graph Quadratic Equations in Two Variables
In the following exercises, graph by using intercepts, the vertex, and the axis of symmetry.
Solve Maximum and Minimum Applications
In the following exercises, find the maximum or minimum value.
In the following exercises, solve. Round answers to the nearest tenth.

### Everyday Math

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

### Chapter 10 Review Exercises

### 10.1 Solve Quadratic Equations Using the Square Root Property

In the following exercises, solve using the Square Root Property.
In the following exercises, solve using the Square Root Property.

### 10.2 Solve Quadratic Equations Using Completing the Square

In the following exercises, complete the square to make a perfect square trinomial. Then write the result as a binomial squared.
In the following exercises, solve by completing the square.

### 10.3 Solve Quadratic Equations Using the Quadratic Formula

In the following exercises, solve by using the Quadratic Formula.
In the following exercises, determine the number of solutions to each quadratic equation.
In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation.

### 10.4 Solve Applications Modeled by Quadratic Equations

In the following exercises, solve by using methods of factoring, the square root principle, or the quadratic formula.

### 10.5 Graphing Quadratic Equations in Two Variables

In the following exercises, graph by plotting point.
In the following exercises, determine if the following parabolas open up or down.
In the following exercises, find ⓐ the axis of symmetry and ⓑ the vertex.
In the following exercises, find the x- and y-intercepts.
In the following exercises, graph by using intercepts, the vertex, and the axis of symmetry.

In the following exercises, find the minimum or maximum value.
In the following exercises, solve. Rounding answers to the nearest tenth.

### Practice Test

Solve the following quadratic equations. Use any method.
Use the discriminant to determine the number of solutions of each quadratic equation.
Solve by factoring, the Square Root Property, or the Quadratic Formula.
For each parabola, find ⓐ which ways it opens, ⓑ the axis of symmetry, ⓒ the vertex, ⓓ the x- and y-intercepts, and ⓔ the maximum or minimum value.
Graph the following parabolas by using intercepts, the vertex, and the axis of symmetry.
Solve.


# Foundations

## Introduction

For years, doctors and engineers have worked to make artificial limbs, such as this hand for people who need them. This particular product is different, however, because it was developed using a 3D printer. As a result, it can be printed much like you print words on a sheet of paper. This makes producing the limb less expensive and faster than conventional methods.
Biomedical engineers are working to develop organs that may one day save lives. Scientists at NASA are designing ways to use 3D printers to build on the moon or Mars. Already, animals are benefitting from 3D-printed parts, including a tortoise shell and a dog leg. Builders have even constructed entire buildings using a 3D printer.
The technology and use of 3D printers depend on the ability to understand the language of algebra. Engineers must be able to translate observations and needs in the natural world to complex mathematical commands that can provide directions to a printer. In this chapter, you will review the language of algebra and take your first steps toward working with algebraic concepts.


# Foundations

## Use the Language of Algebra

In algebra, we use a letter of the alphabet to represent a number whose value may change or is unknown. Commonly used symbols are a, b, c, m, n, x, and y. Further discussion of constants and variables appears later in this section.

### Find Factors, Prime Factorizations, and Least Common Multiples

The numbers 2, 4, 6, 8, 10, 12 are called multiples of 2. A multiple of 2 can be written as the product of 2 and a counting number.
Similarly, a multiple of 3 would be the product of a counting number and 3.
We could find the multiples of any number by continuing this process.
Another way to say that 15 is a multiple of 3 is to say that 15 is divisible by 3. That means that when we divide 15 by 3, we get a counting number. In fact,  is 5, so 15 is
If we were to look for patterns in the multiples of the numbers 2 through 9, we would discover the following divisibility tests:
In mathematics, there are often several ways to talk about the same ideas. So far, we’ve seen that if m is a multiple of n, we can say that m is divisible by n. For example, since 72 is a multiple of 8, we say 72 is divisible by 8. Since 72 is a multiple of 9, we say 72 is divisible by 9. We can express this still another way.
Since  we say that 8 and 9 are factors of 72. When we write  we say we have factored 72.
Other ways to factor 72 are  and  The number 72 has many factors:  and
Some numbers, such as 72, have many factors. Other numbers have only two factors. A prime number is a counting number greater than 1 whose only factors are 1 and itself.
The counting numbers from 2 to 20 are listed in the table with their factors. Make sure to agree with the “prime” or “composite” label for each!
The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19. Notice that the only even prime number is 2.
A composite number can be written as a unique product of primes. This is called the prime factorization of the number. Finding the prime factorization of a composite number will be useful in many topics in this course.
To find the prime factorization of a composite number, find any two factors of the number and use them to create two branches. If a factor is prime, that branch is complete. Circle that prime. Otherwise it is easy to lose track of the prime numbers.
If the factor is not prime, find two factors of the number and continue the process. Once all the branches have circled primes at the end, the factorization is complete. The composite number can now be written as a product of prime numbers.
One of the reasons we look at primes is to use these techniques to find the least common multiple of two numbers. This will be useful when we add and subtract fractions with different denominators.
To find the least common multiple of two numbers we will use the Prime Factors Method. Let’s find the LCM of 12 and 18 using their prime factors.
Notice that the prime factors of 12  and the prime factors of 18  are included in the LCM  So 36 is the least common multiple of 12 and 18.
By matching up the common primes, each common prime factor is used only once. This way you are sure that 36 is the least common multiple.

### Use Variables and Algebraic Symbols

In algebra, we use a letter of the alphabet to represent a number whose value may change. We call this a variable and letters commonly used for variables are
A number whose value always remains the same is called a constant.
To write algebraically, we need some operation symbols as well as numbers and variables. There are several types of symbols we will be using. There are four basic arithmetic operations: addition, subtraction, multiplication, and division. We’ll list the symbols used to indicate these operations below.
When two quantities have the same value, we say they are equal and connect them with an equal sign.
On the number line, the numbers get larger as they go from left to right. The number line can be used to explain the symbols “<” and “>”.
The expressions  or  can be read from left to right or right to left, though in English we usually read from left to right. In general,
Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in English. They help identify an expression, which can be made up of number, a variable, or a combination of numbers and variables using operation symbols. We will introduce three types of grouping symbols now.
Here are some examples of expressions that include grouping symbols. We will simplify expressions like these later in this section.
What is the difference in English between a phrase and a sentence? A phrase expresses a single thought that is incomplete by itself, but a sentence makes a complete statement. A sentence has a subject and a verb. In algebra, we have expressions and equations.
Notice that the English phrases do not form a complete sentence because the phrase does not have a verb.
An equation is two expressions linked by an equal sign. When you read the words the symbols represent in an equation, you have a complete sentence in English. The equal sign gives the verb.
Suppose we need to multiply 2 nine times. We could write this as  This is tedious and it can be hard to keep track of all those 2s, so we use exponents. We write  as  and  as  In expressions such as  the 2 is called the base and the 3 is called the exponent. The exponent tells us how many times we need to multiply the base.
While we read  as  to the  power”, we usually read:
We’ll see later why  and  have special names.
 shows how we read some expressions with exponents.

### Simplify Expressions Using the Order of Operations

To simplify an expression means to do all the math possible. For example, to simplify  we would first multiply  to get 8 and then add the 1 to get 9. A good habit to develop is to work down the page, writing each step of the process below the previous step. The example just described would look like this:
By not using an equal sign when you simplify an expression, you may avoid confusing expressions with equations.
We’ve introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. Otherwise, expressions may have different meanings, and they may result in different values.
For example, consider the expression  Some students simplify this getting 49, by adding  and then multiplying that result by 7. Others get 25, by multiplying  first and then adding 4.
The same expression should give the same result. So mathematicians established some guidelines that are called the order of operations.
Students often ask, “How will I remember the order?” Here is a way to help you remember: Take the first letter of each key word and substitute the silly phrase “Please Excuse My Dear Aunt Sally”.
It’s good that “My Dear” goes together, as this reminds us that multiplication and division have equal priority. We do not always do multiplication before division or always do division before multiplication. We do them in order from left to right.
Similarly, “Aunt Sally” goes together and so reminds us that addition and subtraction also have equal priority and we do them in order from left to right.
When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.

### Evaluate an Expression

In the last few examples, we simplified expressions using the order of operations. Now we’ll evaluate some expressions—again following the order of operations. To evaluate an expression means to find the value of the expression when the variable is replaced by a given number.
To evaluate an expression, substitute that number for the variable in the expression and then simplify the expression.

### Identify and Combine Like Terms

Algebraic expressions are made up of terms. A term is a constant, or the product of a constant and one or more variables.
Examples of terms are  and
The constant that multiplies the variable is called the coefficient.
Think of the coefficient as the number in front of the variable. The coefficient of the term  is 3. When we write  the coefficient is 1, since
Some terms share common traits. When two terms are constants or have the same variable and exponent, we say they are like terms.
Look at the following 6 terms. Which ones seem to have traits in common?
We say,
 and  are like terms.
 and  are like terms.
 and  are like terms.
If there are like terms in an expression, you can simplify the expression by combining the like terms. We add the coefficients and keep the same variable.

### Translate an English Phrase to an Algebraic Expression

We listed many operation symbols that are used in algebra. Now, we will use them to translate English phrases into algebraic expressions. The symbols and variables we’ve talked about will help us do that.  summarizes them.
Look closely at these phrases using the four operations:
Each phrase tells us to operate on two numbers. Look for the words of and and to find the numbers.
We look carefully at the words to help us  distinguish between multiplying a sum and adding a product.
Later in this course, we’ll apply our skills in algebra to solving applications. The first step will be to translate an English phrase to an algebraic expression. We’ll see how to do this in the next two examples.
The expressions in the next example will be used in the typical coin mixture problems we will see soon.

### Key Concepts

1. Divisibility Tests
A number is divisible by:
  2 if the last digit is 0, 2, 4, 6, or 8.
  3 if the sum of the digits is divisible by 3.
  5 if the last digit is 5 or 0.
  6 if it is divisible by both 2 and 3.
  10 if it ends with 0.
2. How to find the prime factorization of a composite number.

3. How To Find the least common multiple using the prime factors method.

4. Equality Symbol
 is read “a is equal to b.”
The symbol “=” is called the equal sign.
5. Inequality

6. Inequality Symbols

7. Grouping Symbols

8. Exponential Notation
 means multiply a by itself, n times.
The expression  is read a to the  power.
9. Simplify an Expression
To simplify an expression, do all operations in the expression.
10. How to use the order of operations.

11. How to combine like terms.

### Practice Makes Perfect

Identify Multiples and Factors
In the following exercises, use the divisibility tests to determine whether each number is divisible by 2, by 3, by 5, by 6, and by 10.
Find Prime Factorizations and Least Common Multiples
In the following exercises, find the prime factorization.
In the following exercises, find the least common multiple of each pair of numbers using the prime factors method.
Simplify Expressions Using the Order of Operations
In the following exercises, simplify each expression.
Evaluate an Expression
In the following exercises, evaluate the following expressions.
Simplify Expressions by Combining Like Terms
In the following exercises, simplify the following expressions by combining like terms.
Translate an English Phrase to an Algebraic Expression
In the following exercises, translate the phrases into algebraic expressions.

### Writing Exercises

### Self Check

ⓐ Use this checklist to evaluate your mastery of the objectives of this section.
ⓑ If most of your checks were:
…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help?Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.


# Foundations

## Integers

### Simplify Expressions with Absolute Value

A negative number is a number less than 0. The negative numbers are to the left of zero on the number line. See .
You may have noticed that, on the number line, the negative numbers are a mirror image of the positive numbers, with zero in the middle. Because the numbers  and  are the same distance from zero, each one is called the opposite of the other. The opposite of  is  and the opposite of  is
 illustrates the definition.
We saw that numbers such as 3 and  are opposites because they are the same distance from 0 on the number line. They are both three units from 0. The distance between 0 and any number on the number line is called the absolute value of that number.
For example,
 illustrates this idea.
The absolute value of a number is never negative because distance cannot be negative. The only number with absolute value equal to zero is the number zero itself because the distance from 0 to 0 on the number line is zero units.
In the next example, we’ll order expressions with absolute values.
We now add absolute value bars to our list of grouping symbols. When we use the order of operations, first we simplify inside the absolute value bars as much as possible, then we take the absolute value of the resulting number.
In the next example, we simplify the expressions inside absolute value bars first just as we do with parentheses.

### Add and Subtract Integers

So far, we have only used the counting numbers and the whole numbers.
Our work with opposites gives us a way to define the integers. The whole numbers and their opposites are called the integers. The integers are the numbers
Most students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more challenging.
We will use two color counters to model addition and subtraction of negatives so that you can visualize the procedures instead of memorizing the rules.
We let one color (blue) represent positive. The other color (red) will represent the negatives.
If we have one positive counter and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero.
We will use the counters to show how to add:
The first example,
 adds 5 positives and 3 positives—both positives.
The second example,  adds 5 negatives and 3 negatives—both negatives.
When the signs are the same, the counters are all the same color, and so we add them. In each case we get 8—either 8 positives or 8 negatives.
So what happens when the signs are different? Let’s add  and
When we use counters to model addition of positive and negative integers, it is easy to see whether there are more positive or more negative counters. So we know whether the sum will be positive or negative.
We will continue to use counters to model the subtraction. Perhaps when you were younger, you read  as “5 take away 3.” When you use counters, you can think of subtraction the same way!
We will use the counters to show to subtract:
The first example,  we subtract 3 positives from 5 positives and end up with 2 positives.
In the second example,  we subtract 3 negatives from 5 negatives and end up with 2 negatives.
Each example used counters of only one color, and the “take away” model of subtraction was easy to apply.
What happens when we have to subtract one positive and one negative number? We’ll need to use both blue and red counters as well as some neutral pairs. If we don’t have the number of counters needed to take away, we add neutral pairs. Adding a neutral pair does not change the value. It is like changing quarters to nickels—the value is the same, but it looks different.
Let’s look at  and
Have you noticed that subtraction of signed numbers can be done by adding the opposite? In the last example,  is the same as  and  is the same as  You will often see this idea, the Subtraction Property, written as follows:
What happens when there are more than three integers? We just use the order of operations as usual.

### Multiply and Divide Integers

Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we are using the model just to help us discover the pattern.
We remember that  means add a, b times.
The next two examples are more interesting. What does it mean to multiply 5 by  It means subtract  times. Looking at subtraction as “taking away”, it means to take away 5, 3 times. But there is nothing to take away, so we start by adding neutral pairs on the workspace.
In summary:
Notice that for multiplication of two signed numbers, when the
What about division? Division is the inverse operation of multiplication. So,  because  In words, this expression says that 15 can be divided into 3 groups of 5 each because adding five three times gives 15. If you look at some examples of multiplying integers, you might figure out the rules for dividing integers.
Division follows the same rules as multiplication with regard to signs.
When we multiply a number by 1, the result is the same number. Each time we multiply a number by  we get its opposite!

### Simplify Expressions with Integers

What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember Please Excuse My Dear Aunt Sally?
Let’s try some examples. We’ll simplify expressions that use all four operations with integers—addition, subtraction, multiplication, and division. Remember to follow the order of operations.
The last example showed us the difference between  and  This distinction is important to prevent future errors. The next example reminds us to multiply and divide in order left to right.

### Evaluate Variable Expressions with Integers

Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.

### Translate Phrases to Expressions with Integers

Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.

### Use Integers in Applications

We’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

### Key Concepts

1. Opposite Notation

2. Absolute Value
The absolute value of a number is its distance from 0 on the number line.
The absolute value of a number n is written as  and  for all numbers.
Absolute values are always greater than or equal to zero.
3. Grouping Symbols

4. Subtraction Property

Subtracting a number is the same as adding its opposite.
5. Multiplication and Division of Signed Numbers
For multiplication and division of two signed numbers:

If the signs are the same, the result is positive.

If the signs are different, the result is negative.
6. Multiplication by

Multiplying a number by  gives its opposite.
7. How to Use Integers in Applications.

### Practice Makes Perfect

Simplify Expressions with Absolute Value
In the following exercises, fill in  or  for each of the following pairs of numbers.
In the following exercises, simplify.
Add and Subtract Integers
In the following exercises, simplify each expression.
Multiply and Divide Integers
In the following exercises, multiply or divide.
Simplify and Evaluate Expressions with Integers
In the following exercises, simplify each expression.
In the following exercises, evaluate each expression.
Translate English Phrases to Algebraic Expressions
In the following exercises, translate to an algebraic expression and simplify if possible.
Use Integers in Applications
In the following exercises, solve.

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?


# Foundations

## Fractions

### Simplify Fractions

A fraction is a way to represent parts of a whole. The fraction  represents two of three equal parts. See . In the fraction  the 2 is called the numerator and the 3 is called the denominator. The line is called the fraction bar.
Fractions that have the same value are equivalent fractions. The Equivalent Fractions
Property allows us to find equivalent fractions and also simplify fractions.
A fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator.
For example,
   is simplified because there are no common factors of 2 and
   is not simplified because 5 is a common factor of 10 and
We simplify, or reduce, a fraction by removing the common factors of the numerator and denominator. A fraction is not simplified until all common factors have been removed. If an expression has fractions, it is not completely simplified until the fractions are simplified.
Sometimes it may not be easy to find common factors of the numerator and denominator. When this happens, a good idea is to factor the numerator and the denominator into prime numbers. Then divide out the common factors using the Equivalent Fractions Property.
We now summarize the steps you should follow to simplify fractions.

### Multiply and Divide Fractions

Many people find multiplying and dividing fractions easier than adding and subtracting fractions.
To multiply fractions, we multiply the numerators and multiply the denominators.
When multiplying fractions, the properties of positive and negative numbers still apply, of course. It is a good idea to determine the sign of the product as the first step. In , we will multiply a negative by a negative, so the product will be positive.
When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, a, can be written as  So, for example,
Now that we know how to multiply fractions, we are almost ready to divide. Before we can do that, we need some vocabulary. The reciprocal of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator. The reciprocal of  is  Since 4 is written in fraction form as  the reciprocal of 4 is
To divide fractions, we multiply the first fraction by the reciprocal of the second.
We need to say   and  to be sure we don’t divide by zero!
The numerators or denominators of some fractions contain fractions themselves. A fraction in which the numerator or the denominator is a fraction is called a complex fraction.
Some examples of complex fractions are:
To simplify a complex fraction, remember that the fraction bar means division. For example, the complex fraction  means

### Add and Subtract Fractions

When we multiplied fractions, we just multiplied the numerators and multiplied the denominators right straight across. To add or subtract fractions, they must have a common denominator.
The least common denominator (LCD) of two fractions is the smallest number that can be used as a common denominator of the fractions. The LCD of the two fractions is the least common multiple (LCM) of their denominators.
After we find the least common denominator of two fractions, we convert the fractions to equivalent fractions with the LCD. Putting these steps together allows us to add and subtract fractions because their denominators will be the same!
We now have all four operations for fractions.  summarizes fraction operations.
When starting an exercise, always identify the operation and then recall the methods needed for that operation.

### Use the Order of Operations to Simplify Fractions

The fraction bar in a fraction acts as grouping symbol. The order of operations then tells us to simplify the numerator and then the denominator. Then we divide.
Where does the negative sign go in a fraction? Usually the negative sign is in front of the fraction, but you will sometimes see a fraction with a negative numerator, or sometimes with a negative denominator. Remember that fractions represent division. When the numerator and denominator have different signs, the quotient is negative.
Now we’ll look at complex fractions where the numerator or denominator contains an expression that can be simplified. So we first must completely simplify the numerator and denominator separately using the order of operations. Then we divide the numerator by the denominator as the fraction bar means division.

### Evaluate Variable Expressions with Fractions

We have evaluated expressions before, but now we can evaluate expressions with fractions. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify.

### Key Concepts

1. Equivalent Fractions Property
If a, b, and c are numbers where  then

2. How to simplify a fraction.

3. Fraction Multiplication
If a, b, c, and d are numbers where  and  then

To multiply fractions, multiply the numerators and multiply the denominators.
4. Fraction Division
If a, b, c, and d are numbers where  and  then

To divide fractions, we multiply the first fraction by the reciprocal of the second.
5. Fraction Addition and Subtraction
If a, b, and c are numbers where  then

To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.
6. How to add or subtract fractions.

7. How to simplify an expression with a fraction bar.

8. Placement of Negative Sign in a Fraction
For any positive numbers a and b,

9. How to simplify complex fractions.

### Practice Makes Perfect

Simplify Fractions
In the following exercises, simplify.
Multiply and Divide Fractions
In the following exercises, perform the indicated operation.
In the following exercises, simplify.
Add and Subtract Fractions
In the following exercises, add or subtract.
Use the Order of Operations to Simplify Fractions
In the following exercises, simplify.
Mixed Practice
In the following exercises, simplify.
Evaluate Variable Expressions with Fractions
In the following exercises, evaluate.

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?


# Foundations

## Decimals

### Round Decimals

Decimals are another way of writing fractions whose denominators are powers of ten.
Just as in whole numbers, each digit of a decimal corresponds to the place value based on the powers of ten.  shows the names of the place values to the left and right of the decimal point.
When we work with decimals, it is often necessary to round the number to the nearest required place value. We summarize the steps for rounding a decimal here.

### Add and Subtract Decimals

To add or subtract decimals, we line up the decimal points. By lining up the decimal points this way, we can add or subtract the corresponding place values. We then add or subtract the numbers as if they were whole numbers and then place the decimal point in the sum.

### Multiply and Divide Decimals

When we multiply signed decimals, first we determine the sign of the product and then multiply as if the numbers were both positive. We multiply the numbers temporarily ignoring the decimal point and then count the number of decimal points in the factors and that sum tells us the number of decimal places in the product. Finally, we write the product with the appropriate sign.
Often, especially in the sciences, you will multiply decimals by powers of 10 (10, 100, 1000, etc). If you multiply a few products on paper, you may notice a pattern relating the number of zeros in the power of 10 to number of decimal places we move the decimal point to the right to get the product.
Just as with multiplication, division of signed decimals is very much like dividing whole numbers. We just have to figure out where the decimal point must be placed and the sign of the quotient. When dividing signed decimals, first determine the sign of the quotient and then divide as if the numbers were both positive. Finally, write the quotient with the appropriate sign.
We review the notation and vocabulary for division:
We’ll write the steps to take when dividing decimals for easy reference.

### Convert Decimals, Fractions, and Percents

In our work, it is often necessary to change the form of a number. We may have to change fractions to decimals or decimals to percent.
We convert decimals into fractions by identifying the place value of the last (farthest right) digit. In the decimal  the 3 is in the hundredths place, so 100 is the denominator of the fraction equivalent to 0.03.
The steps to take to convert a decimal to a fraction are summarized in the procedure box.
A percent is a ratio whose denominator is 100. Percent means per hundred. We use the percent symbol, %, to show percent. Since a percent is a ratio, it can easily be expressed as a fraction. Percent means per 100, so the denominator of the fraction is 100. We then change the fraction to a decimal by dividing the numerator by the denominator. After doing this many times, you may see the pattern.
To convert a percent number to a decimal number, we move the decimal point two places to the left.
To convert a decimal to a percent, remember that percent means per hundred. If we change the decimal to a fraction whose denominator is 100, it is easy to change that fraction to a percent. After many conversions, you may recognize the pattern.
To convert a decimal to a percent, we move the decimal point two places to the right and then add the percent sign.

### Simplify Expressions with Square Roots

Remember that when a number  is multiplied by itself, we write  and read it “ squared.” The result is called the square of a number n. For example,  is read “8 squared” and 64 is called the square of 8. Similarly, 121 is the square of 11 because  is 121. It will be helpful to learn to recognize the perfect square numbers.
What about the squares of negative numbers? We know that when the signs of two numbers are the same, their product is positive. So the square of any negative number is also positive.
Because  we say 100 is the square of 10. We also say that 10 is a square root of 100. A number whose square is m is called a square root of a number m.
Notice  also, so  is also a square root of 100. Therefore, both 10 and  are square roots of 100. So, every positive number has two square roots—one positive and one negative. The radical sign, , denotes the positive square root. The positive square root is called the principal square root. When we use the radical sign that always means we want the principal square root.
We know that every positive number has two square roots and the radical sign indicates the positive one. We write  If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example,  We read  as “the opposite of the principal square root of 10.”

### Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers

We have already described numbers as counting numbers, whole numbers, and integers. What is the difference between these types of numbers? Difference could be confused with subtraction. How about asking how we distinguish between these types of numbers?
What type of numbers would we get if we started with all the integers and then included all the fractions? The numbers we would have form the set of rational numbers. A rational number is a number that can be written as a ratio of two integers.
In general, any decimal that ends after a number of digits (such as 7.3 or ) is a rational number. We can use the reciprocal (or multiplicative inverse) of the place value of the last digit as the denominator when writing the decimal as a fraction. The decimal for  is the number  The bar over the 3 indicates that the number 3 repeats infinitely. Continuously has an important meaning in calculus. The number(s) under the bar is called the repeating block and it repeats continuously.
Since all integers can be written as a fraction whose denominator is 1, the integers (and so also the counting and whole numbers. are rational numbers.
Every rational number can be written both as a ratio of integers  where p and q are integers and  and as a decimal that stops or repeats.
Are there any decimals that do not stop or repeat? Yes! The number  (the Greek letter pi, pronounced “pie”), which is very important in describing circles, has a decimal form that does not stop or repeat. We use three dots (…) to indicate the decimal does not stop or repeat.
The square root of a number that is not a perfect square is a decimal that does not stop or repeat.
A numbers whose decimal form does not stop or repeat cannot be written as a fraction of integers. We call this an irrational number.
Let’s summarize a method we can use to determine whether a number is rational or irrational.
We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. The irrational numbers are numbers whose decimal form does not stop and does not repeat. When we put together the rational numbers and the irrational numbers, we get the set of real numbers.
Later in this course we will introduce numbers beyond the real numbers.  illustrates how the number sets we’ve used so far fit together.
Does the term “real numbers” seem strange to you? Are there any numbers that are not “real,” and, if so, what could they be? Can we simplify  Is there a number whose square is
None of the numbers that we have dealt with so far has a square that is  Why? Any positive number squared is positive. Any negative number squared is positive. So we say there is no real number equal to  The square root of a negative number is not a real number.

### Locate Fractions and Decimals on the Number Line

We now want to include fractions and decimals on the number line. Let’s start with fractions and locate  and  on the number line.
We’ll start with the whole numbers 3 and  because they are the easiest to plot. See .
The proper fractions listed are  and  We know the proper fraction  has value less than one and so would be located between 0 and 1. The denominator is 5, so we divide the unit from 0 to 1 into 5 equal parts  We plot
Similarly,  is between 0 and  After dividing the unit into 5 equal parts we plot
Finally, look at the improper fractions  Locating these points may be easier if you change each of them to a mixed number.
 shows the number line with all the points plotted.
Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.

### Key Concepts

1. How to round decimals.

2. How to add or subtract decimals.

3. How to multiply decimals.

4. How to multiply a decimal by a power of ten.

5. How to divide decimals.

6. How to convert a decimal to a proper fraction and a fraction to a decimal.

7. How to convert a percent to a decimal and a decimal to a percent.

8. Square Root Notation
 is read “the square root of m.”
If  then  for
The square root of m,  is the positive number whose square is m.
9. Rational or Irrational
If the decimal form of a number

10. Real Numbers

### Practice Makes Perfect

Round Decimals
In the following exercises, round each number to the nearest ⓐ hundredth ⓑ tenth ⓒ whole number.
Add and Subtract Decimals
In the following exercises, add or subtract.
Multiply and Divide Decimals
In the following exercises, multiply.
In the following exercises, divide. Round money monetary answers to the nearest cent.
Convert Decimals, Fractions and Percents
In the following exercises, write each decimal as a fraction.
In the following exercises, convert each fraction to a decimal.
In the following exercises, convert each percent to a decimal.
In the following exercises, convert each decimal to a percent.
Simplify Expressions with Square Roots
In the following exercises, simplify.
Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers
In the following exercises, list the ⓐ whole numbers, ⓑ integers, ⓒ rational numbers, ⓓ irrational numbers, ⓔ real numbers for each set of numbers.
Locate Fractions and Decimals on the Number Line
In the following exercises, locate the numbers on a number line.

### Writing Exercises

### Self Check

ⓐ Use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?