8.1 Students will be able to evaluate powers that have zero and negative exponents.

Evaluate the expression.

1. q3 when q = 14

2. c2 when c =35

ANSWER164

ANSWER9

25

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

8.1 Students will be able to evaluate powers that have zero and negative exponents.

Homework Review

8.1 Students will be able to evaluate powers that have zero and negative exponents.

Zero and Negative Exponents

8.1 Students will be able to evaluate powers that have zero and negative exponents.

Notice what occurs when you divide powers with the same base.

5 5 555

53=

5 5 5 5 5= 5 5 = 52=

5 5 55 5 5 5 5

8.1 Students will be able to evaluate powers that have zero and negative exponents.

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=

8.1 Students will be able to evaluate powers that have zero and negative exponents.

d. x61x4

= x6 – 4

= x2

x6

x4=

8.1 Students will be able to evaluate powers that have zero and negative exponents.

Simplify the expression.

1. 611

65

= 66

2. (– 4)9

(– 4)2

= (– 4)7

3. 94 93

92

= 97 – 2

= 95

= 611 – 5

= (– 4)9 – 2

97

92=

8.1 Students will be able to evaluate powers that have zero and negative exponents.

4. y81y5

= y8 – 5

= y3

y8

y5=

8.1 Students will be able to evaluate powers that have zero and negative exponents.EXAMPLE 2

x3

y3 = a. xy

3

(– 7)2

x2=b. 7

x –

2 – 7 x

2= 49

x2=

8.1 Students will be able to evaluate powers that have zero and negative exponents.

Power of a quotient property

Power of a product 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=

8.1 Students will be able to evaluate powers that have zero and negative exponents.

Simplify the expression.

Power of a quotient property

Power of a product property

Power of a power property

= x4

42y2

= x4

16y2

a2

b2=5. ab

2

(– 5)3

y3 = 125 y3–=– 5

y3

=6. = 5y

–3

( x2)2

(4y)2=7. x2

4y2

8.1 Students will be able to evaluate powers that have zero and negative exponents.

Power of a quotient property

Power of a power property

Multiply fractions.

8. 3t

2s 3 t5

1623 s3 33 t3

t5

16=

8 s3 27 t3

t5

16=

8 s3 27 16t3

t5

=

s3 t2

54=

8.1 Students will be able to evaluate powers that have zero and negative exponents.

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.

12

Fractal Tree

8.1 Students will be able to evaluate powers that have zero and negative exponents.EXAMPLE 4 Solve a multi-step problem

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?

8.1 Students will be able to evaluate powers that have zero and negative exponents.EXAMPLE 4 Solve a multi-step problem

a. Step Number of new branches

1 3 = 31

2 9 = 32

3 27 = 33

4 81 = 34

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

8.1 Students will be able to evaluate powers that have zero and negative exponents.GUIDED PRACTICE for Example 4

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 branchadded at Step 9?

12

9.

SOLUTION

( 19 )9 = 512

1units

8.1 Students will be able to evaluate powers that have zero and negative exponents.

ASTRONOMY

EXAMPLE 5Solve a real-world problem

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?

8.1 Students will be able to evaluate powers that have zero and negative exponents.EXAMPLE 5 Solve a real-world

problem

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

1030

10261030 - 26= = 104

ANSWER

Canopus is about 104 times as luminous as the sun.

SOLUTION

8.1 Students will be able to evaluate powers that have zero and negative exponents.GUIDED PRACTICE for Example 5

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?

12

9.

SOLUTION

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

1030

10281030 - 28= = 102

ANSWER

Canopus is about 102 times as luminous as Sirius