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Lesson 1: Exploring Positive and Negative Integer Exponents

By Donna Ventura

Students will use positive and negative integer exponents to generate equivalent numerical expressions for very large and very small numbers.

You are a research biologist studying the life of microorganisms in pond water. In your research study, you observed the number of a certain microorganism doubled each day. You will use integer exponents to record the number of microorganisms each day.

Lesson Objective: Lesson is aligned to the Common Core State Standards for Mathematics – 8.EE.1 Expressions and Equations – Know and apply the properties of integer exponents to generate equivalent numerical expressions.

Materials Required: scientific calculator

Lesson Procedure

Definition: In the number 43, the base number is 4 and the positive integer exponent is 3. The exponent indicates repeated multiplication of the base number. The number 43 means that the base number, 4, will be multiplied by itself 3 times. There are 3 factors of 4 in the number 43. The base number written with a positive integer exponent, 43 is equal to the repeated multiplication sentence 4 x 4 x 4. So 43 = 4 x 4 x 4

1. Look at the number 54.

  • What is the base number?
  • What is the integer exponent? Is it positive or negative?
  • How many factors of 5 are there?
  • Write the repeated multiplication sentence for 54.

2. Look at the repeated multiplication 6 x 6.

  • Write a base number and an integer exponent for the repeated multiplication.
  • Should you use a positive integer exponent or a negative integer exponent?

Answers:

1. Look at the number 54.

  • What is the base number? 5
  • What is the integer exponent? 4. Is it positive or negative? positive
  • How many factors of 5 are there? 4
  • Write the repeated multiplication sentence for 54. 5 x 5 x 5 x 5 = 54

2. Look at the repeated multiplication 6 x 6.

  • Write a base number and an integer exponent for the repeated multiplication. 62
  • Should you use a positive integer exponent or a negative integer exponent? positive

Definition: In the number 2-5, the base number is 2 and the negative integer exponent is -5. A number with a negative exponent is the same as the inverse of the number with a positive exponent. The number 2-5, means that the inverse of the base number, 2, will be multiplied by itself 5 times. There are 5 factors of the inverse of 2 in the number 2-5. The inverse of 2 is 1/2. The base number written with a negative exponent, 2-5 is equal to the inverse written with a positive exponent, 1/25, and equal to the repeated multiplication sentence 1/2 x 1/2 x 1/2 x 1/2 x 1/2. So, 2-5 = 1/25 = 1/2 x 1/2 x 1/2 x 1/2 x 1/2.

3. Look at the number 4-3.

  • What is the base number?
  • What is the integer exponent? Is it positive or negative?
  • How many factors of 4 are there?
  • What is the inverse of 4?
  • Write the repeated multiplication sentence for 4-3.

4. Look at the repeated multiplication 1/6 x 1/6 x 1/6 x 1/6

  • Write a base number and an exponent with a negative integer for the repeated multiplication.
  • Should you use a positive integer exponent or a negative integer exponent?

Answers:

3. Look at the number 4-3.

  • What is the base number? 4
  • What is the integer exponent? -3 Is it positive or negative? Negative
  • How many factors of 4 are there? 3
  • What is the inverse of 4? 1/4
  • Write the repeated multiplication sentence for 4-3. 1/4 x 1/4 x 1/4 = 4-3

4. Look at the repeated multiplication 1/6 x 1/6 x 1/6 x 1/6.

  • Write a base number and an exponent with a negative integer for the repeated multiplication. 6-4
  • Should you use a positive integer exponent or a negative integer exponent? Negative

Individual or Group Work

Download the Student Worksheet

This provides great practice on using integer exponents to generate equivalent numerical expressions for very large and very small numbers.