# Exponents and Exponential Functions

## Objective

Use exponent rules to analyze and rewrite expressions with non-negative exponents.

## 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.

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• 6.EE.A.1

## Criteria for Success

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1. Use the power, product, and quotient rules to simplify exponential expressions with non-negative exponents.
2. Use the order of operations and properties of exponents to write equivalent expressions and to justify why two expressions are not equivalent.

## Tips for Teachers

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This lesson reviews skills and concepts from 8.EE.1 in order to set students up for success with the rest of the unit. 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 expressions are equivalent to ${{(x^5x^4)^3}\over{x^2}}$? Select all that apply.

a.   ${x^{25}}$

b.   ${x^{30}}$

c.   ${{x^{12}}\over{x^2}}$

d.   ${{x^{27}}\over{x^2}}$

e.   ${{x^{60}}\over{x^2}}$

f.   ${{(x^9)^3}\over x^2}$

g.   ${{(x^{20})^3}\over x^2}$

#### Guiding Questions

• Recall the properties of exponents. What are they and what do they state? Where do you see them in action in the problem above?
• What are some common errors or misconceptions that happen when working with exponents? Where do you see examples of these errors in the incorrect answers in the problem above?

### Problem 2

Simplify the following expression using the properties of exponents.

${{{8a^4(7b)^3}\over 7ab^2} \times \left({7^2a}\over{8b^2}\right)^5}$

#### Guiding Questions

• Is there more than one way to approach simplifying this expression? How will you start?
• Where do you see the properties of exponents coming into play?
• What is the implicit exponent when there is not one explicitly shown?

### Problem 3

Is ${(x-y)^2}$ equivalent to ${(x^2-y^2)}$? Justify your answer.

#### Guiding Questions

• How is the expression ${(x-y)^2}$ different from the other expressions you saw in Anchor Problems #1 and #2?
• Are you able to apply the power rule to this expression? Why or why not?
• How will you prove or justify your response?
• Provide a similar situation in which the power rule does not apply.

## 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).

${{3x^3(y^2)^3}\over(2x^2)^3y^4}$