8.2

Warm Up

Lesson Quiz

Lesson Presentation

Apply Exponent Properties Involving Quotients

8.2 Warm-Up

Evaluate the expression.

1. q3 when q = 1 4

2. c2 when c = 3 5

ANSWER 1 64

ANSWER 9 25

8.2 Warm-Up

3. A magazine had a circulation of 9364 in 2001. The circulation was about 125 times greater in 2006. Use order of magnitude to estimate the circulation in 2006.

ANSWER about 106 or 1,000,000

8.2 Example 1

= 86

b. (– 3)9

(– 3)3

= (– 3)6

c. 54 58

57

= 512 – 7

= 55

810

84 a. = 810 – 4 = (– 3)9 – 3

512

57 = d. x6 1 x4

= x6 – 4

= x2

x6

x4 =

Simplify the expression.

8.2 Guided Practice

Simplify the expression.

1. 611

65 = 66

2. (– 4)9

(– 4)2 = (– 4)7

3. 94 93

92 = 95

4. y8 1 y5 = y3

8.2 Example 2

x3

y3 = a. x y

3

(– 7)2

x2 = b. 7

x – 2 – 7

x

2

= 49 x2 =

Simplify the expression.

8.2 Example 3

Power of a quotient property

Power of a product property

= 64x6

125y3 Power of a power property

Power of a quotient property

Power of a power property a10

b5 = 1 2a2

a10

2a2b5 = Multiply fractions.

= 43 (x2)3

53y3

b. a2

b 1 2a2

5

Quotient of powers property a8

2b5 =

(4x2)3

(5y)3 = a. 4x2

5y 3

(a2)5

b5 1 2a2 =

Simplify the expression.

8.2 Guided Practice

= x4

16y2

a2

b2 = 5. a b

2

125 y3 – = 6. 5

y – 3

7. x2

4y 2

8. 3t 2s

3 t5

16 s3 t2

54 =

Simplify the expression.

8.2 Example 4

To construct what is known as a fractal tree, begin with a single segment (the trunk) that is 1 unit long, as in Step 0. Add three shorter segments that are unit long to form the first set of branches, as in Step 1. Then continue adding sets of successively shorter branches so that each new set of branches is half the length of the previous set, as in Steps 2 and 3.

1 2

FRACTAL TREE

8.2 Example 4

a. Make a table showing the number of new branches at each step for Steps 1 - 4. Write the number of new branches as a power of 3.

b. How many times greater is the number of new branches added at Step 5 than the number of new branches added at Step 2?

8.2 Example 4

a.

b. The number of new branches added at Step 5 is 35. The number of new branches added at Step 2 is 32. So, the number of new branches added at Step 5 is = 33 = 27 times the number of new

branches added at Step 2.

35

32

SOLUTION

8.2 Guided Practice

FRACTAL TREE In Example 4, add a column to the table for the length of the new branches at each step. Write the length of the new branches as power of . What is the length of a new branch added at Step 9?

1 2

9.

SOLUTION

( 1 2 )

9 = 512

1 units

8.2 Example 5

ASTRONOMY

The luminosity (in watts) of a star is the total amount of energy emitted from the star per unit of time. The order of magnitude of the luminosity of the sun is 1026 watts. The star Canopus is one of the brightest stars in the sky. The order of magnitude of the luminosity of Canopus is 1030 watts. How many times more luminous is Canopus than the sun?

8.2 Example 5

Luminosity of Canopus (watts) Luminosity of the sun (watts) = 1030

1026 1030 - 26 = = 104

ANSWER

Canopus is about 104 times as luminous as the sun.

SOLUTION

8.2 Guided Practice

9. WHAT IF? Sirius is considered the brightest star in the sky. Sirius is less luminous than Canopus, but Sirius appears to be brighter because it is much closer to the Earth. The order of magnitude of the luminosity of Sirius is 1028 watts. How many times more luminous is Canopus than Sirius?

ANSWER

Canopus is about 102 times as luminous as Sirius.

8.2 Lesson Quiz

ANSWER 64

1. Simplify . 63 65

64

ANSWER 103

ANSWER s24

27r3

3. Simplify s8

3r 3.

–1 10 2. Simplify 107 4.

8.2 Lesson Quiz

ANSWER 103

The order of magnitude of the power output of a nuclear-powered aircraft carrier is about 106 watts. The order of magnitude of peak power at Hoover Dam is about 109 watts. How many times as great is the power output of Hoover Dam as the power output of a nuclear-powered aircraft carrier?

4.