Section 7-2 (and 7-3)Properties of Powers (and Negative Integer Exponents)

Warm-upA multiple choice quiz has two questions. For each

question, there are three choices: A, B, and C. List all possible ways for a student to complete the quiz.

Is the possible number of ways 23 or 32?

Warm-upA multiple choice quiz has two questions. For each

question, there are three choices: A, B, and C. List all possible ways for a student to complete the quiz.

Is the possible number of ways 23 or 32?

(1A, 2A), (1A, 2B), (1A, 2C), (1B, 2A), (1B, 2B), (1B, 2C), (1C, 2A), (1C, 2B), (1C, 2C)

Warm-upA multiple choice quiz has two questions. For each

question, there are three choices: A, B, and C. List all possible ways for a student to complete the quiz.

Is the possible number of ways 23 or 32?

(1A, 2A), (1A, 2B), (1A, 2C), (1B, 2A), (1B, 2B), (1B, 2C), (1C, 2A), (1C, 2B), (1C, 2C)

32

Example 1

32i34

Example 1

32i34

= (3i3)(3i3i3i3)

Example 1

32i34

= (3i3)(3i3i3i3) = 36

Example 1

32i34

= (3i3)(3i3i3i3) = 36

32i34

Example 1

32i34

= (3i3)(3i3i3i3) = 36

32i34

= 32+4

Example 1

32i34

= (3i3)(3i3i3i3) = 36

32i34

= 32+4 = 36

Product of Powers Postulate

bmibn = bm+n

Example 2

(32 )4

Example 2

(32 )4 = (32 )(32 )(32 )(32 )

Example 2

(32 )4 = (32 )(32 )(32 )(32 ) = 38

Example 2

(32 )4 = (32 )(32 )(32 )(32 ) = 38

(32 )4

Example 2

(32 )4 = (32 )(32 )(32 )(32 ) = 38

(32 )4 = 32i4

Example 2

(32 )4 = (32 )(32 )(32 )(32 ) = 38

(32 )4 = 32i4

= 38

Power of a Power Postulate

(bm )n = bmn

Example 3

(x2 y )3

Example 3

(x2 y )3 = (x2 y )(x2 y )(x2 y )

Example 3

(x2 y )3 = (x2 y )(x2 y )(x2 y ) = x6 y3

Example 3

(x2 y )3 = (x2 y )(x2 y )(x2 y ) = x6 y3

(x2 y )3

Example 3

(x2 y )3 = (x2 y )(x2 y )(x2 y ) = x6 y3

(x2 y )3 = (x2 )3 y3

Example 3

(x2 y )3 = (x2 y )(x2 y )(x2 y ) = x6 y3

(x2 y )3 = (x2 )3 y3

= x6 y3

Power of a Product Postulate

(ab)m = ambm

Example 4

(3x2 y3 )4

Example 4

(3x2 y3 )4 = 34(x2 )4 (y3 )4

Example 4

(3x2 y3 )4 = 34(x2 )4 (y3 )4 = 81x8 y12

Example 5

1011

108

Example 5

1011

108 =

(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)

Example 5

1011

108 =

(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)

Example 5

1011

108 =

(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)

= 103

Example 5

1011

108 =

(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)

= 103

1011

108

Example 5

1011

108 =

(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)

= 103

1011

108 = 1011−8

Example 5

1011

108 =

(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)

= 103

1011

108 = 1011−8 = 103

Quotient of Powers Postulate

bm

bn= bm−n

Example 6

x

y

⎛

⎝⎜⎞

⎠⎟

6

Example 6

x

y

⎛

⎝⎜⎞

⎠⎟

6

=

x

y

⎛

⎝⎜⎞

⎠⎟x

y

⎛

⎝⎜⎞

⎠⎟x

y

⎛

⎝⎜⎞

⎠⎟x

y

⎛

⎝⎜⎞

⎠⎟x

y

⎛

⎝⎜⎞

⎠⎟x

y

⎛

⎝⎜⎞

⎠⎟

Example 6

x

y

⎛

⎝⎜⎞

⎠⎟

6

=

x

y

⎛

⎝⎜⎞

⎠⎟x

y

⎛

⎝⎜⎞

⎠⎟x

y

⎛

⎝⎜⎞

⎠⎟x

y

⎛

⎝⎜⎞

⎠⎟x

y

⎛

⎝⎜⎞

⎠⎟x

y

⎛

⎝⎜⎞

⎠⎟ =

x6

y6

Power of a Quotient Postulate

a

b

⎛

⎝⎜⎞

⎠⎟

m

=am

bm

Example 7

x3

x3

Example 7

x3

x3 = x3−3

Example 7

x3

x3 = x3−3 = x0

Example 7

x3

x3 = x3−3 = x0

x3

x3

Example 7

x3

x3 = x3−3 = x0

x3

x3 = 1

Example 7

x3

x3 = x3−3 = x0

x3

x3 = 1

x0 = 1

Zero Exponent Theorem

b0 = 1, b ≠ 0

7-3: Negative Integer Exponents

Example 1

x7

x10

Example 1

x7

x10 = x7−10

Example 1

x7

x10 = x7−10 = x−3

Example 1

x7

x10 = x7−10 = x−3

x7

x10

Example 1

x7

x10 = x7−10 = x−3

x7

x10 =

(x )(x )(x )(x )(x )(x )(x )(x )(x )(x )(x )(x )(x )(x )(x )(x )(x )

Example 1

x7

x10 = x7−10 = x−3

x7

x10 =

(x )(x )(x )(x )(x )(x )(x )(x )(x )(x )(x )(x )(x )(x )(x )(x )(x )

=

1(x )(x )(x )

Example 1

x7

x10 = x7−10 = x−3

x7

x10 =

(x )(x )(x )(x )(x )(x )(x )(x )(x )(x )(x )(x )(x )(x )(x )(x )(x )

=

1(x )(x )(x )

=1

x3

Negative Exponent Theorem

b−n =

1

bn

Example 2

(5b)3 (4b)−5

Example 2

(5b)3 (4b)−5

=

(5b)3

(4b)5

Example 2

(5b)3 (4b)−5

=

(5b)3

(4b)5 =

125b3

1024b5

Example 2

(5b)3 (4b)−5

=

(5b)3

(4b)5 =

125b3

1024b5

=

125

1024b2

Homework

Homework

p. 430 #14-30p. 435 #9-21, 25