Exponents and Exponential Functions

Lesson 5

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).

Foundational Standards

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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

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Problem 2

Write the following expression without negative exponents.

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

Guiding Questions

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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.

  • Include error analysis problems

Target Task

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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.

  $$0<s<1$$ $$-1<s<0$$ $$s\geq1$$ or $$s\leq-1$$
$${x^{-3}}$$      
$$-{x^{-3}}$$      
$${1\over{(-2x)^2}}$$      
$${\left({1\over x}\right)^{-4}}$$