objective :

Objective:

• To apply the properties of exponents.

Properties of Exponents• Power: A power is an expression that

represents repeated multiplication of the same factor. – For example, 81 is a power of 3 because 3x3x3x3

= 81. A power can be written using two numbers, a base and an exponent.

• Exponent: The exponent represents the number of times the base is used as a factor.

• Base: “The big number”

Lesson 8.1

• Apply Exponent Properties Involving Products

Product of Powers Property:

*Power # 1

EXAMPLE 1 Use the product of powers property

= 91 + 8 + 2

= 911

= (– 5)1 + 6

= (–5)7

= 78a. 73 75 = 73 + 5

b. 9 98 92 = 91 98 92

c. (– 5)(– 5)6 = (– 5)1 (– 5)6

d. x4 x3 = x4 + 3 = x7

GUIDED PRACTICE for Example 1

Simplify the expression.

= 391. 32 37

= 510

= (–7)33. (– 7)2(– 7)

4. x2 x6 x = x9

2. 5 59

Power of a Power Property:

Reading Math

(9 ) is read as “nine to the fourth, to the fifth.”

54

*Power # 2

Use the power of a power propertyEXAMPLE 2

= 215 = (–6)10

= x8 = (y + 2)12

a. (25)3

= x2 4c. (x2)4

= (–6)2 5 b. [(–6)2]5

= (y + 2)6 2 d. [(y + 2)6]2

= 25 3


= 414 = (–2)20

= n18 = (m + 1)20

5. (42)7

8. [(m + 1)5]4


6. [(–2)4]5

7. (n3)6

Power of a Product Property:

• To find a power of a product, find the power of each factor and multiply.

*Power # 3

Use the power of a product propertyEXAMPLE 3

d. – (4z)2 = – (4 z)2 = – (42 z2) = –16z2

(9xy)2 = (9 x y)2 = 92 x2 y2 = 81x2y2b.

(–4z)2 = (–4 z)2 = (–4)2 z2 = 16z2c.

a. (24 13)8 = 248 138

Use all three propertiesEXAMPLE 4

Simplify(2x3)2 x4

(2x3)2 x4 = 22 (x3)2 x4

= 4x10

Power of a product property

Power of a power property

Product of powers property

= 4 x6 x4

GUIDED PRACTICE for Examples 3, 4 and 5


(–3n)210.

9. (42 12)2

= 6561m12n411. (9m3n)4

= 422 122

12. 5 (5x2)4 = 3125x8

= 9n2

Lesson 8.2

• Apply Exponent Properties Involving Quotients

Quotient of Powers Property

*Power # 4

EXAMPLE 1 Use the quotient of powers property

= 86

b. (– 3)9

(– 3)3

= (– 3)6

c. 54 58

57

= 512 – 7

= 55

810

84a. = 810 – 4

= (– 3)9 – 3

512

57=

EXAMPLE 1 Use the quotient of powers property

d. x61x4

= x6 – 4

= x2

x6

x4=



1. 611

65 = 66

2. (– 4)9

(– 4)2 = (– 4)7

3. 94 93

92 = 95

4. y81y5 = y3

Power of a Quotient Property

• To find a power of a quotient, find the power of the numerator and the power of the denominator and divide.

*Power # 5

EXAMPLE 2 Use the power of quotient property

x3

y3 = a. xy

3

(– 7)2

x2=b. 7

x –

2 – 7 x

2= 49

x2=

EXAMPLE 3 Use properties of exponents

Power of a quotient property


= 64x6

125y3Power of a power property

Power of a quotient property

Power of a power propertya10

b5= 12a2

a10

2a2b5= Multiply fractions.

= 43 (x2)3

53y3

b. a2

b 12a2

5

Quotient of powers propertya8

2b5=

(4x2)3

(5y)3=a. 4x2

5y3

(a2)5

b512a2=

GUIDED PRACTICE for Examples 2 and 3


= x4

16y2

a2

b2=5. ab

2

125 y3–=6. 5

y –

3

7. x2

4y2

8. 3t

2s 3 t5

16s3 t2

54=

Lesson 8.3

• Define and use Zero and negative exponents

Definition of zero and negative exponents

• Anything to the power of zero is 1 50= 1

• a-n is the reciprocal of an.2-1= ½

• an is the reciprocal of a-n.2= 1/2 -1

*Power # 6

Use definition of zero and negative exponents

EXAMPLE 1

a. 3– 2 Definition of negative exponents

1 9

= Evaluate exponent.

b. (–7)0 Definition of zero exponent

132

=

= 1

Use definition of zero and negative exponents

EXAMPLE 1

= 25 Simplify by multiplying numerator and denominator by 25.

d. 0 – 5 a – n is defined only for a nonzero number a.

10 5

(Undefined)=

=15

1 25

–21c.Definition of negative exponents

125

1=

Evaluate exponent.


Evaluate the expression.

023

1. = 1

1 64

=2. (–8) – 2

12

3.–3

= 8

4. (–1 )0 = 1

Lesson 8.1 – 8.3

• All of the properties of exponents can be used together!

EXAMPLE 2 Evaluate exponential expressions

a. 6– 4 64 Product of a power property

= 60 Add exponents.

= 1 Definition of zero exponent

= 6– 4 + 4


1256

= Evaluate power.

c. 1

3– 4Definition of negative exponents

Evaluate power.= 81

= 34


= 4– 4 Multiply exponents.

Definition of negative exponents

b. (4– 2)2

14

=4

= 4– 2 ∙ 2


1125

= Evaluate power.

d. 5– 1

52Quotient of powers property

= 5– 3 Subtract exponents.

153

= Definition of negative exponents

= 5– 1 – 2


5. 14– 3

= 64

Evaluate the expression.

6. (5– 3) – 1 = 125

7. (– 3 ) (– 3 ) – 55 = 1

8. 6– 2

62

11296

=


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

a. (2xy–5)3 = 23 x3 (y–5)3

= 8 x3 y–15

=y15

8x3





y5

(2x)2(–4x2y2)=(2x)–2y5

–4x2y2b.

y5

(4x)2(–4x2y2)=

y5

–16x4y2=

y3

16x4–=



Product of powers property

Quotient of powers property

objective :

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