1. Exponent Laws II Topic 2.5

2. PowerAs a Repeated Multiplication As a Product of Factors As a PowerAs a Product of Powers (2 4 ) 3 2 4 x 2 4 x 2 4 (2)(2)(2)(2) x 2 12 (3 2 ) 4 [(-4 3 ) 2 ] (2x5) 3 (2x5)x(2x5)x(2x5)2x5x2x5x2x52 3 x 5 3 (3x4) 2 (4x2) 5 OVERVIEW

3. POWER OF A POWER (3 2 ) 4 = 3 2 X 3 2 X 3 2 X 3 2 What do you do with the exponents of like bases when they are multiplied together? (Last section) What do you do with the exponents of like bases when they are multiplied together? (Last section) ADD!!! = 3 2+2+2+2 = 3 8 This answer is the same as multiplying the exponents together. This answer is the same as multiplying the exponents together. =3 2x4

4. POWER OF A POWER Proper Definition (n a ) b = n axb for any n, a, and b in the real numbers.

5. Why don’t we just do this? (3 2 ) 4 = (9) 4 = 9 x 9 x 9 x 9 = 6561 Because sometimes we could get really difficult numbers.

6. Why don’t we just do this? Because sometimes we could get really difficult numbers. (9 12 ) 4 = (282429536481) 4 (282429536481)x(282429536481)x (282429536481)x(282429536481) This is way harder than just doing this: (9 12 ) 4 = 9 12 x 4 = 9 48

7. Exponent Law for POWER OF A POWER To find a power of a power, MULTIPLY the exponents! (6 2 ) 7 = 6 2x7 = [(-7) 3 ] 2 = (-7) 3x2 = -(2 4 ) 5 = -(2 4x5 ) = Write each as a power.

8. POWER OF A PRODUCT =(2x3) 3 =(2x3)(2x3)(2x3) Remember, you can multiply in any order, so group the same numbers =2x2x2x3x3x3 =2 3 x 3 3 Simplify, then evaluate. =216 Is there another way to figure this out? To find a power of product, DISTRIBUTE the exponents to each base!

9. POWER OF A PRODUCT These two methods will give you the same answer. (2x3) 3 =2 3 x 3 3 =216 Method 1Method 2 (2x3) 3 =(6) 3 =216 Again the numbers can get messy on you, and when you start using variables only method 1 will work

10. POWER OF A PRODUCT Proper Definition (m x n) a = m a x n a for any m, n, and a in the real numbers.

11. POWER OF A QUOTIENT To find a power of QUOTIENT, DISTRIBUTE the exponents to each base, then evaluate (if you are asked to!). Simplify First!

12. POWER OF A QUOTIENT Proper Definition for any m, n, and a in the real numbers.

13. Power of a power Power of a product Power of a power (4 3 ) 5 = 4 3x5 = 4 15 (3x8) 4 = 3 4 x 8 4

14. ASSIGNMENT PAGE 84 Page 84-85 #4ace, 5ace, 6ace, 8ace, 13, 14aceg, 16ace, 21