options: Extend the properties of exponents to rational exponents.

Reason quantitatively and use units to solve problems.

Use complex numbers in polynomial identities and equations.

Know that there are numbers that are not rational, and approximate them by rational numbers.

Use properties of rational and irrational numbers.

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Perform operations on matrices and use matrices in applications.

Perform arithmetic operations with complex numbers.

Compute fluently with multi-digit numbers and find common factors and multiples.

Perform operations on vectors.

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Represent complex numbers and their operations on the complex plane.

Apply and extend previous understandings of numbers to the system of rational numbers.

Represent and model with vector quantities.
problem_activity: Activity
The purpose of this activity is for students to extend what they learned about square roots and
\(\frac12\)
exponents to cube roots and
\(\frac13\)
exponents.
Launch
Tell students that they will use the same reasoning as in the previous activity to make sense of numbers to the
\(\frac12\)
power or
\(\frac13\)
power. Give a few minutes of quiet work time followed by a brief, whole-class discussion.
Action and Expression: Internalize Executive Functions.
To support development of organizational skills, check in with students within the first 2–3 minutes of work time. Check that students are finding the exact values and remind them of how they could use the exponent rules to check their answers if needed. Invite students to show their thinking by rewriting the fractional exponents.
Supports accessibility for: Memory; Organization
Student Facing
Use the exponent rules and your understanding of roots to find the exact value of:
\(25^{\frac12}\)
\(15^{\frac12}\)
\(8^{\frac13}\)
\(2^{\frac13}\)
Student Response
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Activity Synthesis
The key takeaway is that extending exponent rules to these fractions means that
\(b^{\frac12}=\sqrt{b}\)
and
\(b^{\frac13}=\sqrt[3]{b}\)
. Select students to share their responses for
\(15^{\frac12}\)
and
\(2^{\frac13}\)
.
Conclude the discussion by reminding students that these expressions which involve the
\(\sqrt{}\)
symbol are often called
radical
expressions because the technical name for the symbol is the
radix
(which is also where the word root comes from). Expressions like
\(\sqrt{15}\)
and
\(\sqrt[3]{2}\)
are examples of radical expressions.
Speaking: MLR8 Discussion Supports.
As students share their responses for
\(15^{\frac{1}{2}}\)
and
\(2^{\frac{1}{3}}\)
, press for details by asking how they know that
\(15^{\frac{1}{2}}\)
is equal to
\(\sqrt{15}\)
and
\(2^{\frac{1}{3}}\)
is equal to
\(\sqrt[3]{2}\)
. Also ask students which properties of exponents they used to show that the exponential and radical expressions are equivalent. Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. This will help students justify why
\(b^{\frac{1}{2}} = \sqrt{b}\)
and
\(b^{\frac{1}{3}} = \sqrt[3]{b}\)
.
Design Principle(s): Support sense-making; Optimize output (for justification)


correct_option_index: 0