prompter: Can you help me understand why and how the distributive law in mathematics works?

assistant: Certainly! The distributive law is a fundamental property of arithmetic that states that multiplying a number by a sum of two or more numbers is the same as multiplying each addend separately and then adding the products. This can be expressed as follows:

a × (b + c) = (a × b) + (a × c)

where a, b, and c are any real numbers.

To understand why the distributive law works, let's consider a concrete example:

3 × (4 + 5)

To evaluate this expression, we can apply the distributive law as follows:

3 × (4 + 5) = (3 × 4) + (3 × 5) = 12 + 15 = 27

We can see that we get the same result as if we added 4 and 5 first and then multiplied the sum by 3:

3 × (4 + 5) = 3 × 9 = 27

The distributive law works because multiplication is distributive over addition, meaning that multiplying a number by a sum is the same as multiplying each addend separately and then adding the products. This property allows us to simplify expressions and perform calculations more efficiently.

The distributive law is also a foundational concept in algebra, where it is used to simplify expressions, factor polynomials, and solve equations. For example, we can use the distributive law to factor the expression 6x + 9y as follows:

6x + 9y = 3(2x + 3y)

In this case, we can see that the distributive law allows us to factor out the greatest common factor of 3 from both terms, which simplifies the expression and makes it easier to work with.

I hope that helps you understand why and how the distributive law works!

assistant: Certainly! The distributive law is a fundamental property of mathematics that allows us to simplify expressions involving multiplication and addition (or subtraction). The law states that:

a × (b + c) = a × b + a × c

In words, this means that when you multiply a number (a) by the sum of two other numbers (b and c), you can distribute the multiplication over the addition, and get the same result as if you had multiplied a by b and then added the result to a times c.

The distributive law works because multiplication is distributive over addition. In other words, if you have two numbers (a and b) and you add them together, and then multiply the result by a third number (c), you can either multiply a by c and add the result to b times c, or you can multiply the sum of a and b by c, and you will get the same result.

To see why this works, let's use some specific numbers. Suppose we want to simplify the expression:

3 × (4 + 5)

Using the distributive law, we can rewrite this as:

3 × 4 + 3 × 5

Which equals:

12 + 15 = 27

We could also have simply multiplied 3 by the sum of 4 and 5 directly, without using the distributive law:

3 × (4 + 5) = 3 × 9 = 27

Either way, we get the same result.

The distributive law is a very useful tool in algebra and other areas of mathematics, because it allows us to simplify complex expressions and equations, and to work with variables and unknowns. For example, we can use the distributive law to expand expressions like:

3(x + 2)

Which equals:

3x + 6

I hope that helps! Let me know if you have any further questions.

assistant: Sure, I can help you with that. The distributive law in mathematics is a law that relates the operations of multiplication and addition. It says that multiplying a number by a group of numbers added together is the same as doing each multiplication separately1. For example:

3 × (2 + 4) = 3×2 + 3×4

This means that you can distribute the factor 3 over the terms inside the parentheses, and then add the products. This is useful when you want to simplify expressions or solve equations.

One way to understand why this works is to think of multiplication as repeated addition. For example, 3 × (2 + 4) means adding 2 + 4 three times:

3 × (2 + 4) = (2 + 4) + (2 + 4) + (2 + 4)

Now, you can use the associative property of addition, which says that you can change the grouping of terms without changing the result. For example:

(2 + 4) + (2 + 4) + (2 + 4) = (2 + 2 + 2) + (4 + 4 + 4)

Then, you can use the commutative property of addition, which says that you can change the order of terms without changing the result. For example:

(2 + 2 + 2) + (4 + 4 + 4) = (3 × 2) + (3 × 4)

This shows that multiplying by a group of numbers added together is equivalent to doing each multiplication separately.