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

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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 - slide 2 of 3
### Lesson Procedure

**Definition:**In the number 4^{3}, the base number is 4 and the positive integer exponent is 3. The exponent indicates repeated multiplication of the base number. The number 4^{3}means that the base number, 4, will be multiplied by itself 3 times. There are 3 factors of 4 in the number 4^{3}. The base number written with a positive integer exponent, 4^{3}is equal to the repeated multiplication sentence 4 x 4 x 4. So 4^{3}= 4 x 4 x 4**1.**Look at the number 5^{4}.- 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 5
^{4}.

**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 5

^{4}.- 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 5
^{4}. 5 x 5 x 5 x 5 = 5^{4}

2. Look at the repeated multiplication 6 x 6.

- Write a base number and an integer exponent for the repeated multiplication. 6
^{2} - 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, 1/2^{5}, and equal to the repeated multiplication sentence 1/2 x 1/2 x 1/2 x 1/2 x 1/2. So, 2^{-5}= 1/2^{5}= 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

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### 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.

#### Working with Radicals and Integer Exponents

- Lesson 1: Exploring Positive and Negative Integer Exponents
- Lesson 2: Exploring Properties of Integer Exponents and Radicals
- Lesson 3: Building Square Patios and Their Roots
- Lesson 4: Building Cube Towers and Their Roots
- Lesson 5: Studying Large Numbers in Space
- Lesson 6: Researching Microorganisms Living in Pond Water