EXAMPLE 1

1. a. (–4 25)2 = 16 252 = 16,384 = 16 210 = (– 4)2(25)2 –1 115 118 = b. 118 115 = 1331 = 113 EXAMPLE 1 Evaluate numerical expressions Power of a product property Power of a power property Simplify and evaluate power. Negative exponent property = 118–5 Quotient of powers property Simplify and evaluate power.

2. = 85,000,000 1200 EXAMPLE 2 Use scientific notation in real life Locusts A swarm of locusts may contain as many as 85 million locusts per square kilometer and cover an area of 1200 squarekilometers. About how many locusts are in such a swarm? SOLUTION Substitute values.

3. = (8.5 107)(1.2 103) = (8.5 1.2)(107 103) = 10.21010 = 1.02 1011010 = 1.02 1011 ANSWER The number of locusts is about1.02 1011, or about102,000,000,000. EXAMPLE 2 Use scientific notation in real life Write in scientific notation. Use multiplication properties. Product of powers property Write 10.2 in scientific notation. Product of powers property

4. 2 3 3. 9 for Examples 1 and 2 GUIDED PRACTICE Evaluate the expression. Tell which properties of exponents you used. 1. (42 )3 4096 ; Power of a power property ANSWER 2.(–8)(–8)3 4096 ; Product of a powers property ANSWER 8 ANSWER ; Power of a quotient property 729

5. 6 10 – 4 9 107 4. for Examples 1 and 2 GUIDED PRACTICE 2 3 1011 ANSWER ; quotient of power property

6. = b–4 +6+7 = b9 r–2–3 ( r –2 )–3 ( s3 )–3 = b. s3 r 6 = s–9 c. 16m4n –5 2n–5 EXAMPLE 3 Simplify expressions Product of powers property a. b–4b6b7 Power of a quotient property Power of a power property = r6s9 Negative exponent property Quotient of powers property = 8m4n –5 – (–5) = 8m4n0= 8m4 Zero exponent property

7. (x–3)2(y3)2 (x–3y3)2 = x5y6 x5y6 x –6y6 = x5y6 EXAMPLE 4 Standardized Test Practice SOLUTION Power of a product property Power of a power property

8. = x–111 1 = x11 The correct answer is B. ANSWER EXAMPLE 4 Standardized Test Practice = x –6– 5y6– 6 Quotient of powers property = x–11y0 Simplify exponents. Zero exponent property Negative exponent property

9. Astronomy Betelgeuse is one of the stars found in the constellation Orion. Its radius is about 1500 times the radius of the sun. How many times as great as the sun’s volume is Betelgeuse’s volume? EXAMPLE 5 Compare real-life volumes

10. 4 π (1500r)3 Betelgeuse’s volume The volume of a sphere is πr3. 3 = 4 Sun’s volume 4 πr3 3 3 4 π15003r3 3 = 4 πr3 3 EXAMPLE 5 Compare real-life volumes SOLUTION Let r represent the sun’s radius. Then 1500r represents Betelgeuse’s radius. Power of a product property

11. = 150031 Betelgeuse’s volume is about 3.4 billiontimes as great as the sun’s volume. ANSWER EXAMPLE 5 Compare real-life volumes = 15003r0 Quotient of powers property Zero exponent property = 3,375,000,000 Evaluate power.

12. 7z4 y2 for Examples 3, 4, and 5 GUIDED PRACTICE Simplify the expression. Tell which properties of exponents you used. 5. x–6x5 x3 x2 ;Product of powers property ANSWER 6. (7y2z5)(y–4z–1) ; Product of powersproperty, Negative exponent property ANSWER

13. x4y–2 3 s 3 2 7. 8. t–4 x3y6 s6t8 x3 y24 for Examples 3, 4, and 5 GUIDED PRACTICE ANSWER ; Power of a power property, Negative exponent property ; Quotient of powers property, Power of a Quotient property, Negative exponent property ANSWER