Exponents and Exponential Functions

Lesson 5


Use negative exponent rules to analyze and rewrite exponential expressions.

Common Core Standards

Core Standards


  • 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


  • 6.EE.A.1

Criteria for Success


  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


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


Problem 1

Which expression is not equivalent to the given expression below?


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.


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


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

Target Task


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 $$s\leq-1$$
$${\left({1\over x}\right)^{-4}}$$