Exponents and Scientific Notation

Lesson 8

Math

Unit 1

8th Grade

Lesson 8 of 15

Objective


Reason with negative exponents to write equivalent, simplified 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.

Criteria for Success


  1. Investigate, determine, and apply the general rule for negative exponents: $${{x^{-m}} = {1\over{x^{m}}}}$$  and $${{1\over{{x^{-m}}}}={x^m}}$$; understand $${x^{-m}}$$ as the reciprocal of $${x^m}$$.
  2. Simplify exponential expressions using exponent rules.

Tips for Teachers


A common error when working with negative exponents is to over-apply the rule to bases that have an implicit exponent of 1 or to negative numbers (not exponents). For example, $${2x^{-1}}$$ is often incorrectly written as $${1\over{2x}}$$ , not realizing that the base 2 has a positive exponent of 1 and should remain in the numerator.

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Anchor Problems


Problem 1

What is the value of $${{{{{5^{-2}}}}}}$$? Explore 3 different strategies to determine the value. 

Strategy 1: Use product of powers rule

How does the statement below shed light on the value of $${{{{{5^{-2}}}}}}$$?

$${5^2}\cdot{{{{{5^{-2}}}}}}=5^{2+(-2)}={5^0}=1$$

Strategy 2: Use quotient of power rule

How does the statement below shed light on the value of $${{{{{5^{-2}}}}}}$$?

$${5^4\over5^6}={{5\cdot5\cdot5\cdot5}\over{5\cdot5\cdot5\cdot5\cdot5\cdot5}}=5^{4-6}={{{{{5^{-2}}}}}}$$

Strategy 3: Use a table

Complete the table below to shed light on the value of $${{{{{5^{-2}}}}}}$$.

Exponent Value
$${5^3}$$  
$${5^2}$$  
$${5^1}$$  
$${5^0}$$  
$${5^{-1}}$$  
$${{{{{5^{-2}}}}}}$$  

Guiding Questions

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Problem 2

Simplify the following expressions to the fewest number of bases possible and no negative exponents.

a.   $${15^4\over{15^{-3}}}$$

b.   $${2x^{-5}}$$

c.   $${(ab^2)^{-3}}$$

d.   $${m^{-3}\over m}$$

e.   $${\left ({1\over3} \right )^{-2}}$$

f.   $${1\over{7^{-8}}}$$

Guiding Questions

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Problem Set

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Target Task


The expressions in each example below are not equivalent. Explain a possible mistake that could have been made and then find an equivalent expression.

a.   $${{x^{-3}y^2\over{y^{-4}x^2}}={1\over{y^6x^5}}}$$

b.   $${4m^{-1}={1\over{4m}}}$$

Student Response

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Additional Practice


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

  • Problems similar to: $${x^8=x^{10}\cdot x^?}$$
  • Problems similar to: Which is larger? $${2^5\over{2^{-5}}}$$ OR $${2^{-5}\over 2^5}$$?
  • Challenge: For what values of $$a$$ is $$a^{-b}>a^b$$, when $$b$$ is a positive integer?
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Lesson 7

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Lesson 9

Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Review of Exponents

Topic B: Properties of Exponents

Topic C: Scientific Notation

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