# Exponents and Exponential Functions

## Objective

Multiply and divide rational exponent expressions and radical expressions.

## Common Core Standards

### Core Standards

?

• N.RN.A.2 — Rewrite expressions involving radicals and rational exponents using the properties of exponents.

• N.RN.B.3 — Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

?

• 8.EE.A.1

• 8.EE.A.2

• 8.NS.A.1

## Criteria for Success

?

1. Understand that ${{\sqrt[n]{a}}\times\sqrt[n]{b}=\sqrt[n]{ab}}$, and that ${{{{\sqrt[n]{a}}}\over{\sqrt[n]{b}}}=\sqrt[n]{a\over b}}$.
2. Understand that ${\sqrt[n]{a}}$ and ${\sqrt[m]{a}}$ can be rewritten as ${a^{1\over n}}$ and ${a^{1\over m}}$ in order to be multiplied or divided.
3. Apply the properties of exponents and properties of operations to multiply and divide rational exponent and radical expressions.
4. Understand that a rational number multiplied by a rational number is rational, and a rational number multiplied by an irrational number is irrational.

## Anchor Problems

?

### Problem 1

All of the following equations are true.

${\sqrt{16}\cdot\sqrt9=\sqrt{144}}$              ${{\sqrt{80}\over\sqrt{4}}=\sqrt{20}}$               ${\sqrt5\cdot\sqrt[3]{5}=\sqrt[6]{5^5}}$

a.   What general rules can you determine from these examples?

b.   Find the products or quotients below

i.   ${\sqrt{12}\cdot\sqrt2}$

ii.  ${\sqrt[3]{4}\cdot\sqrt[3]{3}\cdot\sqrt[3]{5}}$

iii.   ${\sqrt2\cdot\sqrt3\cdot\sqrt[3]{6}}$

iv.   ${\sqrt[3]{45}\over\sqrt[3]{5}}$

#### Guiding Questions

• What do you notice happens when you multiply two square roots?
• What do you notice happens when you divide two square roots?
• How can you express these occurrences as a general rule?
• What if the radicals have different indices (e.g., square root and cube root)? Can you multiply the numbers under the radicals (the radicands)?
• When is it useful to change a radical expression into a rational exponent expression?

### Problem 2

Multiply and simplify as much as possible.

a.   ${-5\sqrt{12}\cdot\sqrt8}$

b.   ${\sqrt[3]{6x^2}\cdot\sqrt[3]{9x^4}}$

Divide and simplify as much as possible.

c.   ${2\sqrt{6}\div\sqrt{24}}$

d.   ${\sqrt{120m^9}\over{10\sqrt{4m^4}}}$

#### Guiding Questions

• When is it the best approach to multiply the radicands (expressions under the radicals)?
• When is it the best approach to change the radicals into rational exponent expressions?
• Can you simplify the expression ${\sqrt{5}\over5}$? Why or why not? Can you simplify the expression ${\sqrt{5}\over\sqrt5}$? Why or why not?
• Are either of your factors irrational? Is your product irrational?
• Is either the divisor or dividend irrational? Is your quotient irrational?

### Problem 3

Compute and simplify.

a.   ${\sqrt{10}\cdot2\sqrt[3]{10}}$

b.   ${3\sqrt[3]{16}\div2\sqrt[3]{54}}$

#### Guiding Questions

• When is it the best approach to multiply the radicands (expressions under the radicals)?
• When is it the best approach to change the radicals into rational exponent expressions?
• Are either of your factors irrational? Is your product irrational?
• Is either the divisor or dividend irrational? Is your quotient irrational?

#### Notes

This Anchor Problem is optional and can be moved to the problem set depending on the timing of your class and/or needs of your students.

## Problem Set

?

The following resources include problems and activities aligned to the objective of the lesson. They can be used to create a problem set for class (for non-Fishtank Plus users), or as supplementary or additional resources to the pre-made Problem Set (for Fishtank Plus users).

## Target Task

?

Find the error in each solution. Then find the correct product or quotient.

a.   ${\sqrt{20}\cdot\sqrt[3]{5}=\sqrt{100}=10}$

b.   ${{{4\sqrt{35}}\over{\sqrt{28}}}={{4\sqrt{5\cdot7}}\over{\sqrt{4\cdot7}}}={\sqrt5}}$