Evaluate the expression. 1. q3 when q = 1 4 ANSWER 1 64 2. c2 when c = 3 5 ANSWER 9 25

3. A magazine had a circulation of 9364 in 2001 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

EXAMPLE 1 Use the quotient of powers property 810 84 a. = 810 – 4 = 86 b. (– 3)9 (– 3)3 = (– 3)9 – 3 = (– 3)6 512 57 = c. 54 58 57 = 512 – 7 = 55

EXAMPLE 1 Use the quotient of powers property d. x6 1 x4 x6 x4 = = x6 – 4 = x2

GUIDED PRACTICE for Example 1 Simplify the expression. 1. 611 65 = 66 2. (– 4)9 (– 4)2 = (– 4)7 3. 94 93 92 = 95 4. y8 1 y5 = y3

EXAMPLE 2 Use the power of quotient property a. x y 3 x3 y3 = b. 7 x – 2 – 7 x 2 = (– 7)2 x2 = 49 x2 =

Use properties of exponents EXAMPLE 3 Use properties of exponents a. 4x2 5y 3 (4x2)3 (5y)3 = Power of a quotient property = 43 (x2)3 53y3 Power of a product property = 64x6 125y3 Power of a power property b. a2 b 1 2a2 5 (a2)5 b5 1 2a2 = Power of a quotient property a10 b5 = 1 2a2 Power of a power property a10 2a2b5 = Multiply fractions. a8 2b5 = Quotient of powers property

GUIDED PRACTICE for Examples 2 and 3 Simplify the expression. 5. a b 2 a2 b2 = 6. 5 y – 3 125 y3 – = 7. x2 4y 2 = x4 16y2 8. 3t 2s 3 t5 16 s3 t2 54 =

EXAMPLE 4 Solve a multi-step problem FRACTAL TREE 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

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?

EXAMPLE 4 Solve a multi-step problem SOLUTION 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

( ) GUIDED PRACTICE for Example 4 9. FRACTAL TREE In Example 4, add a column to the table for the length of the new branches at each step. Write the lengths of the new branches as powers of . What is the length of a new branch added at Step 9? 1 2 9. SOLUTION ( 1 2 ) 9 = 512 units

EXAMPLE 5 Solve a real-world problem 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?

EXAMPLE 5 Solve a real-world problem SOLUTION 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.

GUIDED PRACTICE for Example 5 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.

Daily Homework Quiz 1. Simplify . 63 65 64 ANSWER 64 –1 10 2. Simplify 107 4. ANSWER 103 3. Simplify s8 3r 3. ANSWER s24 27r3

Daily Homework Quiz 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. ANSWER 103