1. 6.1 Properties of Exponents 1/8/2014

2. Power, Base and Exponent: 7373 Exponent: is the number that tells you how many times the base is multiplied to itself. In this example 7 3 means 777

3. Product of Powers: Ex. 3 2 3 5 = 3333333 = 3 7 = 3 2+5 In general: a ma n = a m+n Note: bases must be the same.

4. Power of a Power: Ex. (2 3 ) 2 = (2 3 ) (2 3 ) =(222)(222) = 2 6 = 2 32 In general: (a m ) n = a mn

5. Power of a Product: Ex. (4x ) 3 = (4x ) (4x ) (4x ) =(444)(xxx) = 4 3 x 3 In general: (a b) m =a m b m Note: Distribute the outside exponent to the inside exponent.

6. Zero Exponent In general: a 0 = 1 ÷5 ÷4 ÷3 Any base raised to a 0 power equals 1. 5 0 = 1 4 0 = 1 3 0 = 1

7. What happens when a base is raised to a negative power? The power moves from top to bottom and the exponent becomes positive!

8. Negative Exponent Ex. 5 -2 Ex. In general: Negative exponent MOVES power. If the power with a negative exponent is in the numerator, the power moves to the denominator and exponent becomes positive. If the power with a negative exponent is in the denominator, the power moves to the numerator and exponent becomes positive.

9. Quotient of Powers Ex. In general: Note: bases must be the same.

10. Power of a Quotient Ex. In general:

11. Example 1 Evaluate Expressions with Negative Exponents () 8 2 – – ()4)4 2 – Since bases are the same, add exponents = () 8 4 2 – – + = () 4 2 – – 1 ()4)4 2 – = Move the power with negative exponent = 16 1 Evaluate power.

12. Example 2 Evaluate Quotients with Exponents 3 3535 2 Evaluate. Since bases are the same, subtract exponents 3 3535 2 = ()2)2 3232 = 3434 = 81 Evaluate power.

13. Checkpoint Evaluate Numerical Expressions ()3)32 1. Evaluate the expression. ANSWER 64 ()3)3 5050 2. ANSWER 1 3. () 5 3 – – ()2)2 3 – ANSWER 27 1 – ANSWER 27 8 4. 3 2 3

14. Example 3 Simplify Algebraic Expressions a. y x – 3 2 Distribute the outside exponents x 2x 2 ()2)2 y – 3 == x 2x 2 y – 6 Simplify exponent. = x 2y 6x 2y 6 Move the power with negative exponent

15. Example 3 Simplify Algebraic Expressions = 25y y 5yy 5y – 6 Simplify exponent. = 25y – 6 5 1 ++ Add exponents of powers with same base. = 25y 0 = 25 Apply Zero exponent prop ()2)2 5y5y – 3 y 5yy 5y b. = ()2)2 y – 3 y 5yy 5y 5252 Distribute outside exponent

16. Example 3 Simplify Algebraic Expressions c. x 5 y 2 – x 3y 6x 3y 6 = y 8y 8 x 2x 2 To simplify the powers of x, think: which power has more x and by how many? Move the power with negative exponent The bottom x has more by 2.

17. Checkpoint Simplify Algebraic Expressions ()3)3 2p2pp 4p 4 5. 8p 78p 7 6. xy 4 – x 5y 3x 5y 3 x 4y 7x 4y 7 Simplify the expression. 7. ()3)3 3b3b – 2 b 8b 8 27b 2 – s r 2r 2 – 4 3 8. 1 r 6 s 12 ANSWER

18. Write an expression in simplified form for the Volume of the cylinder. r = radius h = height

19. Homework: 6.1 p.299 #8-34 even, 40, 42 Disregard direction that says. “Tell which properties you used”