Lesson 8.3 Define and Use Zero and Negative Exponents

After today’s lesson, you should be able to use zero and negative to

simplify expressions. (CA 2.0)

Power Value 1

1

1

1

1

1

1

1

Zero Exponent

Ex 1) Simplify Method 1: Use Expanded Form

Zero Exponent

Method 2: Use the Quotient of Powers Rule

Definition: Zero exponent

Words: a to the zero power is 1.

Algebra:

Examples: Simplify.

2)

Examples: Simplify.

3)

Examples: Simplify.

4) You Try:

Power Value

Negative Exponents

Using Patterns to Discover the Meaning Behind Negative Exponents

Complete the following table. Use only a base of 10.Stand. Form Expanded Form Expon. Form

100,000

10,000

1000

100

10

1

Complete the following table. Use only a base of 10.Stand. Form Expanded Form Expon. Form

100,000 10 10 10 10 1∙ ∙ ∙ ∙0

10,000

1000

100

10

1

Complete the following table. Use only a base of 10.Stand. Form Expanded Form Expon. Form

100,000 10 10 10 10 1∙ ∙ ∙ ∙0

105

10,000

1000

100

10

1

Complete the following table. Use only a base of 10.Stand. Form Expanded Form Expon. Form

100,000 10 10 10 10 1∙ ∙ ∙ ∙0

105

10,000 10 10 10 10∙ ∙ ∙

1000

100

10

1

Complete the following table. Use only a base of 10.Stand. Form Expanded Form Expon. Form

100,000 10 10 10 10 1∙ ∙ ∙ ∙0

105

10,000 10 10 10 10∙ ∙ ∙ 104

1000

100

10

1

Complete the following table. Use only a base of 10.Stand. Form Expanded Form Expon. Form

100,000 10 10 10 10 1∙ ∙ ∙ ∙0

105

10,000 10 10 10 10∙ ∙ ∙ 104

1000 10 10 10∙ ∙

100

10

1

Complete the following table. Use only a base of 10.Stand. Form Expanded Form Expon. Form

100,000 10 10 10 10 1∙ ∙ ∙ ∙0

105

10,000 10 10 10 10∙ ∙ ∙ 104

1000 10 10 10∙ ∙ 103

100

10

1

Complete the following table. Use only a base of 10.Stand. Form Expanded Form Expon. Form

100,000 10 10 10 10 1∙ ∙ ∙ ∙0

105

10,000 10 10 10 10∙ ∙ ∙ 104

1000 10 10 10∙ ∙ 103

100 10 10∙

10

1

Complete the following table. Use only a base of 10.Stand. Form Expanded Form Expon. Form

100,000 10 10 10 10 1∙ ∙ ∙ ∙0

105

10,000 10 10 10 10∙ ∙ ∙ 104

1000 10 10 10∙ ∙ 103

100 10 10∙ 102

10

1

Complete the following table. Use only a base of 10.Stand. Form Expanded Form Expon. Form

100,000 10 10 10 10 1∙ ∙ ∙ ∙0

105

10,000 10 10 10 10∙ ∙ ∙ 104

1000 10 10 10∙ ∙ 103

100 10 10∙ 102

10 10

1

Complete the following table. Use only a base of 10.Stand. Form Expanded Form Expon. Form

100,000 10 10 10 10 1∙ ∙ ∙ ∙0

105

10,000 10 10 10 10∙ ∙ ∙ 104

1000 10 10 10∙ ∙ 103

100 10 10∙ 102

10 10 101

1

Complete the following table. Use only a base of 10.Stand. Form Expanded Form Expon. Form

100,000 10 10 10 10 1∙ ∙ ∙ ∙0

105

10,000 10 10 10 10∙ ∙ ∙ 104

1000 10 10 10∙ ∙ 103

100 10 10∙ 102

10 10 101

1 100

Now let’s continue the pattern.Stand. Form Expanded Form Expon. Form

100,000 10 10 10 10 1∙ ∙ ∙ ∙0

105

10,000 10 10 10 10∙ ∙ ∙ 104

1000 10 10 10∙ ∙ 103

100 10 10∙ 102

10 10 101

1 100

110

1100

11000

110,000

Now let’s continue the pattern.Stand. Form Expanded Form Expon. Form

100,000 10 10 10 10 1∙ ∙ ∙ ∙0

105

10,000 10 10 10 10∙ ∙ ∙ 104

1000 10 10 10∙ ∙ 103

100 10 10∙ 102

10 10 101

1 100

110

110

1100

11000

110,000

Now let’s continue the pattern.Stand. Form Expanded Form Expon. Form

100,000 10 10 10 10 1∙ ∙ ∙ ∙0

105

10,000 10 10 10 10∙ ∙ ∙ 104

1000 10 10 10∙ ∙ 103

100 10 10∙ 102

10 10 101

1 100

110

110 10-1

1100

11000

110,000

Now let’s continue the pattern.Stand. Form Expanded Form Expon. Form

100,000 10 10 10 10 1∙ ∙ ∙ ∙0

105

10,000 10 10 10 10∙ ∙ ∙ 104

1000 10 10 10∙ ∙ 103

100 10 10∙ 102

10 10 101

1 100

110

110 10-1

1100

110 10∙

11000

110,000

Now let’s continue the pattern.Stand. Form Expanded Form Expon. Form

100,000 10 10 10 10 1∙ ∙ ∙ ∙0

105

10,000 10 10 10 10∙ ∙ ∙ 104

1000 10 10 10∙ ∙ 103

100 10 10∙ 102

10 10 101

1 100

110

110 10-1

1100

110 10∙ 10-2

11000

110,000

Now let’s continue the pattern.Stand. Form Expanded Form Expon. Form

100,000 10 10 10 10 1∙ ∙ ∙ ∙0

105

10,000 10 10 10 10∙ ∙ ∙ 104

1000 10 10 10∙ ∙ 103

100 10 10∙ 102

10 10 101

1 100

110

110 10-1

1100

110 10∙ 10-2

11000

110 10 10∙ ∙

110,000

Now let’s continue the pattern.Stand. Form Expanded Form Expon. Form

100,000 10 10 10 10 1∙ ∙ ∙ ∙0

105

10,000 10 10 10 10∙ ∙ ∙ 104

1000 10 10 10∙ ∙ 103

100 10 10∙ 102

10 10 101

1 100

110

110 10-1

1100

110 10∙ 10-2

11000

110 10 10∙ ∙ 10-3

110,000

Now let’s continue the pattern.Stand. Form Expanded Form Expon. Form

100,000 10 10 10 10 1∙ ∙ ∙ ∙0

105

10,000 10 10 10 10∙ ∙ ∙ 104

1000 10 10 10∙ ∙ 103

100 10 10∙ 102

10 10 101

1 100

110

110 10-1

1100

110 10∙ 10-2

11000

110 10 10∙ ∙ 10-3

110,000

110 10 10 10∙ ∙ ∙

Now let’s continue the pattern.Stand. Form Expanded Form Expon. Form

100,000 10 10 10 10 1∙ ∙ ∙ ∙0

105

10,000 10 10 10 10∙ ∙ ∙ 104

1000 10 10 10∙ ∙ 103

100 10 10∙ 102

10 10 101

1 100

110

110 10-1

1100

110 10∙ 10-2

11000

110 10 10∙ ∙ 10-3

110,000

110 10 10 10∙ ∙ ∙ 10-4

What should you learn from this?Stand. Form Expanded Form Expon. Form

100,000 10 10 10 10 1∙ ∙ ∙ ∙0

105

10,000 10 10 10 10∙ ∙ ∙ 104

1000 10 10 10∙ ∙ 103

100 10 10∙ 102

10 10 101

1 100

110

110 10-1

1100

110 10∙ 10-2

11000

110 10 10∙ ∙ 10-3

110,000

110 10 10 10∙ ∙ ∙ 10-4

What should you learn from this?Stand. Form Expanded Form Expon. Form

100,000 10 10 10 10 1∙ ∙ ∙ ∙0

105

10,000 10 10 10 10∙ ∙ ∙ 104

1000 10 10 10∙ ∙ 103

100 10 10∙ 102

10 10 101

1 100

110

110 10-1

1100

110 10∙ 10-2

11000

110 10 10∙ ∙ 10-3

110,000

110 10 10 10∙ ∙ ∙ 10-4

1) Any number raised to the zero power is always 1.

What should you learn from this?Stand. Form Expanded Form Expon. Form

100,000 10 10 10 10 1∙ ∙ ∙ ∙0

105

10,000 10 10 10 10∙ ∙ ∙ 104

1000 10 10 10∙ ∙ 103

100 10 10∙ 102

10 10 101

1 100

110

110 10-1

1100

110 10∙ 10-2

11000

110 10 10∙ ∙ 10-3

110,000

110 10 10 10∙ ∙ ∙ 10-4

1) Any number raised to the zero power is always 1.

2) A negative exponent will create a fraction.

A negative exponent will result in the reciprocal of the base with a positive exponent.

A negative exponent will result in the reciprocal of the base with a positive exponent.

5-2 =

A negative exponent will result in the reciprocal of the base with a positive exponent.

5-2 = 15( )

2

A negative exponent will result in the reciprocal of the base with a positive exponent.

5-2 = 15( )

2

= 15∙

15

= 125

A negative exponent will result in the reciprocal of the base with a positive exponent.

4-3 =

A negative exponent will result in the reciprocal of the base with a positive exponent.

4-3 = 14( )

3

A negative exponent will result in the reciprocal of the base with a positive exponent.

4-3 = 14( )

3

= 14∙

14

= 164∙1

4

A negative exponent will result in the reciprocal of the base with a positive exponent.

3-4 =

A negative exponent will result in the reciprocal of the base with a positive exponent.

3-4 = 13( )

4

A negative exponent will result in the reciprocal of the base with a positive exponent.

3-4 = 13( )

4

= 13∙

13

= 181∙1

3 ∙13

A negative exponent will result in the reciprocal of the base with a positive exponent.

(-6)-3 =

( )

A negative exponent will result in the reciprocal of the base with a positive exponent.

= 1-6

3

(-6)-3

( )

A negative exponent will result in the reciprocal of the base with a positive exponent.

= 1-6

3

= 1-6∙

1-6

= 1216∙1

-6-(-6)-3

Practice Time

Evaluate.

1) 2-3 = 2) 4-4 =

3) (-2)-4 = 4) (-5)-3 =

5) 80 = 6) (-4)0 =

Evaluate.

1) 2-3 = 2) 4-4 =

3) (-2)-4 = 4) (-5)-3 =

5) 80 = 6) (-4)0 =

18

Evaluate.

1) 2-3 = 2) 4-4 =

3) (-2)-4 = 4) (-5)-3 =

5) 80 = 6) (-4)0 =

18

1256

Evaluate.

1) 2-3 = 2) 4-4 =

3) (-2)-4 = 4) (-5)-3 =

5) 80 = 6) (-4)0 =

18

1256

116

Evaluate.

1) 2-3 = 2) 4-4 =

3) (-2)-4 = 4) (-5)-3 =

5) 80 = 6) (-4)0 =

18

1256

116

1125-

Evaluate.

1) 2-3 = 2) 4-4 =

3) (-2)-4 = 4) (-5)-3 =

5) 80 = 6) (-4)0 =

18

1256

116

1125-

1

Evaluate.

1) 2-3 = 2) 4-4 =

3) (-2)-4 = 4) (-5)-3 =

5) 80 = 6) (-4)0 =

18

1256

116

1125-

1 1

Evaluate. Remember the order of operations.

1) 4-2 + 2-3 =

2) 50 – (6)-2 =

3) 4(3 + 5)0 + 42 =

4) (3 + 5)-2 =

Evaluate. Remember the order of operations.

1) 4-2 + 2-3 =

2) 50 – (6)-2 =

3) 4(3 + 5)0 + 42 =

4) (3 + 5)-2 =

316

Evaluate. Remember the order of operations.

1) 4-2 + 2-3 =

2) 50 – (6)-2 =

3) 4(3 + 5)0 + 42 =

4) (3 + 5)-2 =

316

3536

Evaluate. Remember the order of operations.

1) 4-2 + 2-3 =

2) 50 – (6)-2 =

3) 4(3 + 5)0 + 42 = 20

4) (3 + 5)-2 =

316

3536

Evaluate. Remember the order of operations.

1) 4-2 + 2-3 =

2) 50 – (6)-2 =

3) 4(3 + 5)0 + 42 = 20

4) (3 + 5)-2 =

316

164

3536

Evaluate. Remember the order of operations.

5) 2-1 + (-2)-2 =

6) 52 – 30 =

7) 2(1 + 3)-2 =

8) (6 + 3)2 + (6 + 3)-2 =

Evaluate. Remember the order of operations.

5) 2-1 + (-2)-2 =

6) 52 – 30 =

7) 2(1 + 3)-2 =

8) (6 + 3)2 + (6 + 3)-2 =

34

Evaluate. Remember the order of operations.

5) 2-1 + (-2)-2 =

6) 52 – 30 =

7) 2(1 + 3)-2 =

8) (6 + 3)2 + (6 + 3)-2 =

34

24

Evaluate. Remember the order of operations.

5) 2-1 + (-2)-2 =

6) 52 – 30 =

7) 2(1 + 3)-2 =

8) (6 + 3)2 + (6 + 3)-2 =

34

24

18

Evaluate. Remember the order of operations.

5) 2-1 + (-2)-2 =

6) 52 – 30 =

7) 2(1 + 3)-2 =

8) (6 + 3)2 + (6 + 3)-2 = 81

34

24

18

181

Definition: Negative exponent

Words:

Algebra:

Examples: Evaluate the expression.

5)

Examples: Evaluate the expression.

6)

Examples: Evaluate the expression.

7)

=49

Examples: Evaluate the expression.

8) You Try

Examples: Evaluate the expression.

9)

Examples: Evaluate the expression.

10)

Examples: Evaluate the expression.11) You Try

Examples: Evaluate the expression.12) You Try

Examples: Simplify the expression. Write your answer using only positive exponents.

13)

Examples: Simplify the expression. Write your answer using only positive exponents.

14)

Examples: Simplify the expression. Write your answer using only positive exponents.

15)

Examples: Simplify the expression. Write your answer using only positive exponents.

16) You Try

Examples: Simplify the expression. Write your answer using only positive exponents.

17)

Examples: Simplify the expression. Write your answer using only positive exponents.

18)

Examples: Simplify the expression. Write your answer using only positive exponents.

19) You Try