# Exponents and Exponential Functions

## Objective

Use negative exponent rules to analyze and rewrite exponential expressions.

## Common Core Standards

### Core Standards

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• 8.EE.A.1 — Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3² × 3-5 = 3-3 = 1/3³ = 1/27.

• A.SSE.A.2 — Use the structure of an expression to identify ways to rewrite it. For example, see x4 — y4 as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).

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• 6.EE.A.1

## Criteria for Success

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1. Understand that ${x^{-m}={1\over x^m}}$ and ${{1\over x^{-m}}=x^m}$.
2. Use properties of exponents to simplify expressions including negative and zero exponents.
3. Analyze the structure of an exponential expression and determine an efficient way to write a simplified equivalent expression (Standard for Mathematical Practice 7).

## Tips for Teachers

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This lesson reviews skills and concepts from 8.EE.1. Depending on the needs of your students, this lesson may be skipped or used in a different way.

## Anchor Problems

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### Problem 1

Which expression is not equivalent to the given expression below?

${{x^{-2}}\over{y^{-3}}}$

a.   ${x^{-2}y^3}$

b.   ${{y^3}\over{x^2}}$

c.   ${1\over{x^2y^{-3}}}$

d.   ${x^{-2}\cdot{1\over y^3}}$

e.   ${y^3\cdot{1\over x^2}}$

#### Guiding Questions

• How is ${x^{-2}}$ written with a positive exponent?
• How is ${1\over y^{-3}}$ written with a positive exponent?
• Give one reason to explain why ${3^{-2}}$ is equal to ${{1\over3^2}}$.
• Write each expression above without negative exponents.

### Problem 2

Write the following expression without negative exponents.

${(2x^{-2}3y^2)^{-1}\over{x^5y^{-2}}}$

#### Guiding Questions

• Describe the structure of the exponential expression.
• What is your strategy for simplifying this expression? Where will you start?
• Compare your approach with a peer. Did you get the same simplified expression?

## Problem Set

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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).

• Include error analysis problems

Given that ${x>1}$ and $s$ represents the value of the expression, put a check mark in the appropriate column to indicate the value, $s$, of each expression.
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