# Lesson 2: Exploring Properties of Integer Exponents and Radicals

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

In this lesson, you will use properties of integer exponents to generate equivalent numerical expressions for very large and very small numbers.

**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

In this lesson we will explore the rules (properties) for integer exponents.

**Rule 1:** = *a ^{n }b^{n}=(a x b)^{n}*

Look at the expression on the left side of the equation. Are the exponents the same number? When the exponents are the same number in a multiplication problem, multiply the base numbers; multiply *a* and *b*. The product of *a* x *b* is then raised to the exponent, *n*.

Example 1

- Let’s try it together: What does 6
^{5}x 3^{5}equal? - Are the exponents the same?
- What is the product of 6 times 3?
- The product is raised to what exponent?
- What does 6
^{5}x 3^{5}equal?

Answers

- Let’s try it together: What does 6
^{5}x 3^{5}equal? - Are the exponents the same? Yes
- What is the product of 6 times 3? 18
- The product is raised to what exponent? 5
- What does 6
^{5}x 3^{5}equal? (6 x 3)^{5}= 18^{5}= 1,889,568

Example 2

- Let’s try it again by yourself: 7
^{4}x 2^{4} - Are the exponents the same?
- What is the product of 7 times 2?
- The product is raised to what exponent?
- What does 7
^{4}x 2^{4}equal?

Answers

- Let’s try it again by yourself: 7
^{4}x 2^{4} - Are the exponents the same? Yes
- What is the product of 7 times 2? 14
- The product is raised to what exponent? 4
- What does 7
^{4}x 2^{4}equal? (7 X 2)^{4}= 14^{2}= 38,416

Example 3

- One more time: What does 4
^{3}x 2^{3 }equal?

Answer

- 4
^{3}x 2^{3}= (4 x 2)^{3}= 8^{3}= 512

**Rule 2:** a^{n}/b^{n} = (a/b)^{n}

Look at the expression on the left side of the equation. Are the exponents the same number? When the exponents are the same number in a division problem, divide the base numbers; *a* divided by *b*. The quotient of *a* divided by *b* is then raised to the exponent, *n*.

Example 4

- Let’s try it together: What does 8
^{5}/4^{5}equal? - Are the exponents the same?
- What is the quotient of 8 divided by 4?
- The quotient is raised to what exponent?
- What does 8
^{5}/4^{5}equal?

Answers

- Let’s try it together: What does 8
^{5}/4^{5}equal? - Are the exponents the same? Yes
- What is the quotient of 8 divided by 4? 2
- The quotient is raised to what exponent? 5
- What does 8
^{5}/4^{5}equal? (8/5)^{5}= 2^{5}= 32

Example 5

- Let’s try it again by yourself: 14
^{2}/7^{2} - Are the exponents the same?
- What is the quotient of 14 divided by 7?
- The quotient is raised to what exponent?
- What does 14
^{2}/7^{2}equal?

Answers

- Let’s try it together: 14
^{2}/7^{2} - Are the exponents the same? Yes
- What is the quotient of 14 divided by 7? 2
- The quotient is raised to what exponent? 2
- What does 14
^{2}/7^{2}equal? (14/7)^{2}= 2^{2}= 4

Example 6

- One more time: What does 63
^{2/}7^{2 }equal?

Answer

- 63
^{2/}7^{2 }= (63/7)^{2}= 9^{2}= 81

**Rule 3:** (a^{n})^{m}=a^{nxm}

Look at the expression on the left side of the equation. The base number is raised to a power and then raised to another power. When the base and the exponent are raised to another exponent, multiply the exponents.

Example 7

- Let’s try it together: What does (7
^{2})^{3}equal? - Are the base and the exponent raised to another exponent?
- What is the product of the exponents, 2 x 3?
- The base is raised to what exponent?
- What does (7
^{2})^{3}equal?

Answers

- Let’s try it together: What does (7
^{2})^{3}equal? - Are the base and the exponent raised to another exponent? Yes
- What is the product of the exponents, 2 x 3? 6
- The base is raised to what exponent? 6
- What does (7
^{2})^{3}equal? (7^{2})^{3}= 7^{2x3}= 7^{6}= 117,649

Example 8

- Let’s try it again by yourself: (5
^{3})^{4} - Are the base and the exponent raised to another exponent?
- What is the product of the exponents, 3 x 4?
- The base is raised to what exponent?
- What does (5
^{3})^{4}equal?

Answers

- Let’s try it again by yourself: (5
^{3})^{4} - Are the base and the exponent raised to another exponent? Yes
- What is the product of the exponents, 3 x 4? 12
- The base is raised to what exponent? 12
- What does (5
^{3})^{4}equal? (5^{3})^{4}= 5^{3x4}= 5^{12}= 244,140,625

Example 9

- One more time: What does (8
^{5})^{2}equal?

Answer

- (8
^{5})^{2}= 8^{5x2 }= 8^{10}= 1,073,741,824

**Rule 4:** a^{n }a^{m }= a^{n+m}

Look at the expression on the left side of the equation. Are the exponents the same number? Are the bases the same number? When the bases are the same number in a multiplication problem, add the exponent numbers; add *n* and *m*. The base number, *a*, is then raised to the sum of the exponents, *n* + *m*.

Example 10

- Let’s try it together: What does 9
^{2}9^{3}equal? - Are the bases the same number?
- What is the sum of the exponents, 2 + 3?
- The base is raised to what exponent?
- What does 9
^{2}9^{3}equal?

Answers

- Let’s try it together: What does 9
^{2}9^{3}equal? - Are the bases the same number? Yes
- What is the sum of the exponents, 2 + 3? 5
- The base is raised to what exponent? 5
- What does 9
^{2}9^{3}equal? 9^{2}9^{3}= 9^{2+3}= 9^{5}= 59,049

Example 11

- Let’s try it again by yourself: 3
^{5}3^{4} - Are the bases the same number?
- What is the sum of the exponents?
- The base is raised to what exponent?
- What does 3
^{5}3^{4}equal?

Answers

- Let’s try it again by yourself: 3
^{5}3^{4} - Are the bases the same number? Yes
- What is the sum of the exponents? 9
- The base is raised to what exponent? 9
- What does 3
^{5}3^{4}equal? 3^{5}3^{4}= 3^{5+4}= 3^{9}= 19,683

Example 12

- One more time: What does 8
^{2}8^{4 }equal?

Answer

- 8
^{2}8^{4}= 8^{2+4}= 8^{6}= 262,144

**Rule 5: **a^{b}/a^{m} = a^{n-m}

Look at the expression on the left side of the equation. Are the exponents the same number? Are the bases the same number? When the bases are the same number in a division problem, subtract the exponent numbers; subtract *n* and *m*. The base number, *a*, is then raised to the difference of the exponents, *n* – *m*.

Example 13

- Let’s try it together: What does 5
^{6}/5^{2}equal? - Are the bases the same number?
- What is the difference of the exponents, 6 - 2?
- The base is raised to what exponent?
- What does 5
^{6}/5^{2}equal?

Answers

- Let’s try it together: What does 5
^{6}/5^{2}equal? - Are the bases the same number? Yes
- What is the difference of the exponents, 6 - 2? 4
- The base is raised to what exponent? 4
- What does 5
^{6}/5^{2}equal? 5^{6}/5^{2}= 5^{6-2}= 5^{4}= 625

Example 14

- Let’s try it again by yourself: 4
^{8}/4^{5} - Are the bases the same number?
- What is the difference of the exponents?
- The base is raised to what exponent?
- What does 4
^{8}/4^{5}equal?

Answers

- Let’s try it together: What does 4
^{8}/4^{5}equal? - Are the bases the same number? Yes
- What is the difference of the exponents? 3 – 5 = -2
- The base is raised to what exponent? -2
- What does 4
^{8}/4^{5}equal? 4^{8}/4^{5}= 4^{3-5}= 4^{-2}= 1/4^{2}= 1/16

Example 15

- One more time: What does 7
^{8}/7^{5}equal?

Answer:

- 7
^{8}/7^{5}= 7^{8-5}= 7^{3}= 343