3. Example 1 Evaluate Expressions with Negative Exponents () 8 2 – – ()4)4 2 – Product of powers property = () 8 4 2 – – + = () 4 2 – – Simplify exponent. 1 ()4)4 2 – = Negative exponent property = 16 1 Evaluate power.

4. Example 2 Evaluate Quotients with Exponents 3 3535 2 Evaluate. Quotient of powers property 3 3535 2 = ()2)2 3232 = 3434 Power of a power property = 81 Evaluate power.

5. Checkpoint Evaluate Numerical Expressions ()3)3 2424 1. Evaluate the expression. ANSWER 4096 ()3)3 5050 2. ANSWER 1 3. () 7 3 – – ()2)2 3 – ANSWER 243 1 – ANSWER 343 8 4. 7 2 3

6. Example 3 Simplify Algebraic Expressions a. y x – 3 2 Power of a quotient property x 2x 2 ()2)2 y – 3 = ()2)2 5y5y – 3 y 5yy 5y b. = x 2x 2 y – 3 2 Power of a power property = x 2x 2 y – 6 Simplify exponent. = x 2y 6x 2y 6 Negative exponent property = ()2)2 y – 3 y 5yy 5y 5252 Power of a product property = 25y y 5yy 5y – 3 2 Power of a power property

7. Example 3 Simplify Algebraic Expressions = 25y y 5yy 5y – 6 c. x 5 y 2 – x 3y 6x 3y 6 Simplify exponent. = 25y – 6 5 1 ++ Product of powers property = 25y 0 Simplify exponent. = 25 Zero exponent property = x – 3 5 y – 6 ( 2) – Quotient of powers property = x 2 y 8 – Simplify exponent. = y 8y 8 x 2x 2 Negative exponent property

8. Checkpoint Simplify Algebraic Expressions ()3)3 2p2pp 4p 4 5. ANSWER ; power of a product, product of powers 8p 78p 7 6. xy 4 – x 5y 3x 5y 3 ANSWER ; quotient of powers x 4y 7x 4y 7 Simplify the expression. Tell which properties of exponents you used.

9. Checkpoint Simplify Algebraic Expressions 7. ()3)3 3b3b – 2 b 8b 8 ANSWER ; power of a product, power of a power, product of powers 27b 2 Simplify the expression. Tell which properties of exponents you used.

10. Checkpoint Simplify Algebraic Expressions – s r 2r 2 – 4 3 8. ANSWER 1 r 6 s 12 ; power of a quotient, power of a power, negative exponent Simplify the expression. Tell which properties of exponents you used.

11. DRUMS The two heads of a bass drum are hit using mallets to produce a low pitch. The bass drum used by Purdue University’s marching band has a diameter of about 8 feet and a depth of about 4 feet.

12. Example 4 Compare Areas Drums The head of the bass drum used by Purdue University’s marching band has a radius about 4 times greater than a common bass drum. How many times greater is the area of Purdue’s bass drum’s head than a common bass drum’s head? Let r represent the radius of a common bass drum’s head. SOLUTION Divide the area of Purdue’s drum head by the area of a common drum head. Area of common drum head Area of Purdue’s drum head πr 2πr 2 ()2)2 4r4r π = The area of a circle is A r 2. π =

13. Example 4 Compare Areas = πr 2πr 2 π4242 r 2r 2 Power of a product property = 4242 π 1 1 – r 2 2 – Quotient of powers property = 4242 π0π0 r 0r 0 Simplify exponents. = 16 1 1 = 16 Zero exponent property The area of Purdue’s bass drum’s head is about 16 times greater. ANSWER

14. Example 5 Use Scientific Notation Transportation In 2003 the population of the United States was 2.91 10 8. In that same year, the total motor fuel usage was 1.70 10 11 gallons. Estimate the average number of gallons used per person in the United States in 2003. SOLUTION Divide the total motor fuel usage by the population. × × Population Total fuel usage = × 10 11 ×1.70 10 8 2.91 Divide fuel used by population. = × 1.70 2.91 10 11 8 – Quotient of powers property

15. Example 5 Use Scientific Notation = × 1.70 2.91 10 3 Simplify exponent. ×10 3 0.584≈ Use a calculator. = 584 Simplify. The average number of gallons used per person was about 584. ANSWER

16. Checkpoint Use Scientific Notation 9. In 2003, about 4.25 10 11 local phone calls were made in the United States. Use the population in Example 5 to estimate the average number of local calls made per person. ANSWER about 1460 local calls ×

21. LAKE BAIKAL is the largest freshwater lake by volume, and the Caspian Sea is the world’s largest saltwater lake by volume. The map at the right shows how their areas compare.