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Evaluate the expression.

1. 1. q 3 when q =. 4. 3. ANSWER. 1. 9. 5. 64. 25. about 10 6 or 1,000,000. ANSWER. 2. c 2 when c =. ANSWER. Warm-Up. Evaluate the expression.

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Evaluate the expression.

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  1. 1 1.q3whenq = 4 3 ANSWER 1 9 5 64 25 about 106or 1,000,000 ANSWER 2.c2whenc = ANSWER Warm-Up Evaluate the expression. 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.

  2. Homework Review

  3. Zero and Negative Exponents

  4. 5 5 5555 5 5555 5 53 = = = 5  5 = 52 555 555 Notice what occurs when you divide powers with the same base.

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

  6. x6 d. x6 = x4 1 x4 = x6 – 4 = x2

  7. 1. (– 4)9 2. (– 4)2 611 97 65 92 94 93 92 3. = Simplify the expression. = 611 – 5 = 66 = (– 4)9 – 2 = (– 4)7 = 97 – 2 = 95

  8. y8 y5 4. y8 = 1 y5 = y8 – 5 = y3

  9. 3 a. = – 7 (– 7)2 49 x x2 x2 7 x3 x 2 2 y3 x y – = = = b. EXAMPLE 2

  10. 3 4x2 (4x2)3 64x6 a. = a8 5y (5y)3 125y3 = 2b5 43 (x2)3 = 53y3 = 1 a10 1 a2 (a2)5 2a2 b5 2a2 1 b 5 b. = 2a2 b5 = a10 2a2b5 = Power of a quotient property Power of a product property Power of a power property Power of a quotient property Power of a power property Multiply fractions. Quotient of powers property

  11. x2 ( x2)2 4y (4y)2 2 5. = – 5 (–5)3 125 y y3 y3 3 3 – – = = = 6. = a2 5 a b b2 y 2 = 7. x4 x4 = = 42y2 16y2 Simplify the expression. Power of a quotient property Power of a product property Power of a power property

  12. 8 s3 23s3 t5 t5 8. = 33t3 27 t3 16 16 3t 16 = 8 s3 t5 3 = t5 2s 27 16t3 s3 t2 54 = Power of a quotient property Power of a power property Multiply fractions.

  13. 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

  14. 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. How many times greater is the number of new branches added at Step 5 than the number of new branches added at Step 2? b. EXAMPLE 4 Solve a multi-step problem

  15. 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 EXAMPLE 4 Solve a multi-step problem SOLUTION a.

  16. 9. 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 1 ( ) 2 9 = 9 1 units 512 for Example 4 GUIDED PRACTICE SOLUTION

  17. 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 ASTRONOMY

  18. Luminosity of Canopus (watts) 1030 1030 - 26 104 = = = Luminosity of the sun (watts) 1026 ANSWER Canopus is about 104 times as luminous as the sun. EXAMPLE 5 Solve a real-world problem SOLUTION

  19. 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? Luminosity of Canopus (watts) 1030 1030 - 28 102 = = = Luminosity of the sun (watts) 1028 1 2 ANSWER Canopus is about 102 times as luminous as Sirius for Example 5 GUIDED PRACTICE SOLUTION

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