# Exponents and Exponential Functions

## Objective

Multiply and divide rational exponent expressions and radical expressions.

## Common Core Standards

### Core Standards

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• 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.

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• 8.EE.A.1

• 8.EE.A.2

• 8.NS.A.1

## Criteria for Success

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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

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

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

### Problem 3

Compute and simplify.

a.   ${\sqrt{10}\cdot2\sqrt[3]{10}}$

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

## Problem Set

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The following resources include problems and activities aligned to the objective of the lesson that can be used to create your own problem set.

a.   ${\sqrt{20}\cdot\sqrt[3]{5}=\sqrt{100}=10}$
b.   ${{{4\sqrt{35}}\over{\sqrt{28}}}={{4\sqrt{5\cdot7}}\over{\sqrt{4\cdot7}}}={\sqrt5}}$