# Exploring Properties of Integer Exponents and Radicals: 8th Grade Math Lesson

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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: = an bn=(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 65 x 35 equal?
• Are the exponents the same?
• What is the product of 6 times 3?
• The product is raised to what exponent?
• What does 65 x 35 equal?

• Let’s try it together: What does 65 x 35 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 65 x 35 equal? (6 x 3)5 = 185 = 1,889,568

Example 2

• Let’s try it again by yourself: 74 x 24
• Are the exponents the same?
• What is the product of 7 times 2?
• The product is raised to what exponent?
• What does 74 x 24 equal?

• Let’s try it again by yourself: 74 x 24
• 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 74 x 24 equal? (7 X 2)4 = 142 = 38,416

Example 3

• One more time: What does 43 x 23 equal?

• 43 x 23 = (4 x 2)3 = 83 = 512

Rule 2: an/bn = (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 85/45 equal?
• Are the exponents the same?
• What is the quotient of 8 divided by 4?
• The quotient is raised to what exponent?
• What does 85/45 equal?

• Let’s try it together: What does 85/45 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 85/45 equal? (8/5)5 = 25 = 32

Example 5

• Let’s try it again by yourself: 142/72
• Are the exponents the same?
• What is the quotient of 14 divided by 7?
• The quotient is raised to what exponent?
• What does 142/72 equal?

• Let’s try it together: 142/72
• 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 142/72 equal? (14/7)2 = 22 = 4

Example 6

• One more time: What does 632/72 equal?

• 632/72 = (63/7)2 = 92 = 81

Rule 3: (an)m=anxm

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 (72)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 (72)3 equal?

• Let’s try it together: What does (72)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 (72)3 equal? (72)3 = 72x3 = 76 = 117,649

Example 8

• Let’s try it again by yourself: (53)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 (53)4 equal?

• Let’s try it again by yourself: (53)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 (53)4 equal? (53)4 = 53x4 = 512 = 244,140,625

Example 9

• One more time: What does (85)2 equal?

• (85)2 = 85x2 = 810 = 1,073,741,824

Rule 4: an am = an+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 9293 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 9293 equal?

• Let’s try it together: What does 9293 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 9293 equal? 9293 = 92+3 = 95 = 59,049

Example 11

• Let’s try it again by yourself: 3534
• Are the bases the same number?
• What is the sum of the exponents?
• The base is raised to what exponent?
• What does 3534 equal?

• Let’s try it again by yourself: 3534
• 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 3534 equal? 3534 = 35+4 = 39 = 19,683

Example 12

• One more time: What does 8284 equal?

• 8284 = 82+4 = 86 = 262,144

Rule 5: ab/am = an-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, nm.

Example 13

• Let’s try it together: What does 56/52 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 56/52 equal?

• Let’s try it together: What does 56/52 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 56/52 equal? 56/52 = 56-2 = 54 = 625

Example 14

• Let’s try it again by yourself: 48/45
• Are the bases the same number?
• What is the difference of the exponents?
• The base is raised to what exponent?
• What does 48/45 equal?

• Let’s try it together: What does 48/45 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 48/45 equal? 48/45 = 43-5 = 4-2 = 1/42 = 1/16

Example 15

• One more time: What does 78/75 equal?