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8.1 Students will be able to evaluate powers that have zero and negative exponents. Evaluate the expression. 1 1. q3 when q = 4 ANSWER 2. c2 when c = 3 5 ANSWER 1 64 9 25 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. 55555 55555 55 2 = 5 5 = = = 5 555 555 53 8.1 Students will be able to evaluate powers that have zero and negative exponents. a. 810 = 810 – 4 84 = 86 b. (– 3)9 = (– 3)9 – 3 (– 3)3 = (– 3)6 c. 12 54 58 = 5 57 57 = 512 – 7 = 55 Use the quotient of powers property 8.1 Students will be able to evaluate powers that have zero and negative exponents. d. x6 1 6 x = 4 x x4 = x6 – 4 = x2 8.1 Students will be able to evaluate powers that have zero and negative exponents. Simplify the expression. 1. 611 11 – 5 =6 65 = 66 2. (– 4)9 9–2 = (– 4) 2 (– 4) = (– 4)7 3. 94 93 = 97 92 92 = 97 – 2 = 95 8.1 Students will be able to evaluate powers that have zero and negative exponents. 4. y8 1 8 y = 5 y y5 = y8 – 5 = y3 8.1 Students will be able to evaluate powers that have zero and EXAMPLE 2 negative exponents. a. b. x 3 x3 y = y3 –7 x 2 = –7 x 2 (– 7)2 49 = = x2 x2 8.1 Students will be able to evaluate powers that have zero and negative exponents. 4x2 5y a. b. 5 2 a b 3 (4x2)3 = (5y)3 43 (x2)3 = 3 3 5y 64x6 = 125y3 (a2)5 1 2a2 = b5 a10 = 5 b a10 = 2a2b5 8 = a 2b5 1 2a2 1 2a2 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 8.1 Students will be able to evaluate powers that have zero and negative exponents. Simplify the expression. 5. a 2 a2 = 2 b b 5 6. = – y 7. x2 4y 3 –5 = y 2 3 2)2 ( x = (4y)2 x4 = 2 2 4y x4 = 16y2 (– 5)3 125 = 3 =– y y3 Power of a quotient property Power of a product property Power of a power property 8.1 Students will be able to evaluate powers that have zero and negative exponents. 8. 2s 3t 3 t5 16 23 s3 = 3 3 3 t t5 16 8 s3 t 5 = 27 t3 16 8 s3 t5 = 27 16t3 s3 t 2 = 54 Power of a quotient property Power of a power property Multiply fractions. 8.1 Students will be able to evaluate powers that have zero and negative exponents. 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 1 unit 2 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. 8.1 Students will be able to evaluate powers that have zero and EXAMPLE 4 Solve a multi-step problem negative exponents. 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 EXAMPLE 4 Solve a multi-step problem negative exponents. SOLUTION a. Step Number of new branches 1 3 = 31 b. 2 9 = 32 3 27 = 33 4 81 = 34 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 5 Step 5 is 3 = 33 = 27 times the number of new 32 branches added at Step 2. 8.1 Students will be able to evaluate powers that have zero and for Example 4 GUIDED PRACTICE negative exponents. 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 1 . What is the length of a new branch 2 added at Step 9? SOLUTION 1 1 9 ( 9 ) = 512 units 8.1 Students will be able to evaluate powers that have zero and EXAMPLE 5 Solve a real-world problem negative exponents. 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? 8.1 Students will be able to evaluate powers that have zero and EXAMPLE 5 Solve a real-world problem negative exponents. SOLUTION Luminosity of Canopus (watts) 1030 30 - 26 4 10 10 = = = Luminosity of the sun (watts) 1026 ANSWER Canopus is about 104 times as luminous as the sun. 8.1 Students will be able to evaluate powers that have zero and for Example 5 GUIDED PRACTICE negative exponents. 9. WHAT IF? Sirius is considered the brightest star in the sky. Sirius is less luminous than Canopus, but Sirius 1 appears to be brighter because it is much closer to the 2 Earth. The order of magnitude of the luminosity of Sirius is 1028 watts. How many times more luminous is Canopus than Sirius? SOLUTION Luminosity of Canopus (watts) 1030 30 - 28 2 10 10 = = = Luminosity of the sun (watts) 1028 ANSWER Canopus is about 102 times as luminous as Sirius