1. Reviewing the Exponent Laws

2. Throughout the study of all modern sciences, extremely large and extremely small numbers frequently appear.

3. The distance from the Earth to the Sun 150 000 000 000 m

4. 1 000 000 000 000 000 Number of cells in the human body

5. An iPad can hold up to 32 000 000 000 bytes of information

6. In an attempt to be more efficient when operating on these numbers, a series of mathematical shortcuts were derived. These shortcuts evolved into the Exponent Laws

7. The Exponent Laws are one set of tools a mathematician can use to help make work quicker and easier.

8. 1. Multipliying Powers Simplify the following powers (click to see the answer) 1. 2 3 = 2 X 2 X 2 = 8 2. 5 2 = 5 X 5 = 25 NOTE: 2525 base exponent

9. Now we try this!!! 6. 2 4 X 2 3 We expand each expression = 2 X 2 X 2 X 2 X 2 X 2 X 2 How many 2s are you multiplying together? 7 = 2 7

10. Notice the exponents 3. 2 4 X 2 3 Is there a way would could get 7 given the initial exponents of 4 and 3 = 2 7 Correct! You can add the exponents together This property holds true for multiplying powers with the same base.

11. Express as a single power. (click to see each answer) 3. 2 5 X 2 2 = 4. 3 2 X 3 4 = 5. 1 7 X 1 3 = 2727 3 6 1 10

12. ExLaw #1 Let B be any base Let x and y be any exponent (B x )(B y ) = B x + y “When multiplying powers with the same base, add the exponents!!!”

13. For example: (x 3 )(x 8 ) =x 11 (a 4 )(a 3 ) =a7a7

14. Consider the Division of powers 2525 2323 Expand each power = 2 X 2 X 2 X 2 X 2 2 X 2 X 2 Notice: There are now numbers on the top and the bottom that can be divided out!!! 1 1 1 1 1 1 = 2 X 2 = 2 2

15. Notice the exponents: 2525 2323 = Is there anything you could do with a 5 and a 3 to get 2? 2 2 Subtract! That is correct! This is true for any division of powers with the same base

16. Reduce the following to a single power 2727 2323 = 1. 2 7 - 3 = 2 4 4343 4141 = 2. 4 3 - 1 = 4 2 5656 5252 = 3. 5 6 - 2 = 5 4

17. ExLaw #2 Let B be any base Let x and y be any exponent (B x ) = B x - y (B y ) “When dividing powers with the same base, subtract the exponents!!!”

18. 3. Power of Powers Expand the following: Sometimes the base you are expanding is a power itself! 2 3 =2 X 2 X 2

19. Expand the following: Expand this in the same way (2 2 ) 3 2 2 X 2 2 X 2 2 = = 2 X 2 X 2 X 2 X 2 X 2 Which can be written as … = 2 6 How many 2s are you multiplying?... 6

20. Examine the exponents (2 2 ) 3 = 2 6 What can you do with 2 and 3 to get 6? Multiply! Correct! This property is true for all power of powers with the same base.

21. Reduce the following to a single power (2 7 ) 2 = 1. 2 7 X 2 = 2 14

22. ExLaw #3 Let B be any base Let x and y be any exponent (B x ) y = B (x X y) “When expanding a power of powers, multiply the exponents!!!”

23. To understand the fourth and fifth exponent laws, examine the following pattern

24. ExLaw #4 x 0 = 1 2 3 =8 2 2 =4 2 1 = 2 Continue the pattern 2 0 = 1 2 x 1 2 x 1 2 x 1 “Any base to the exponent zero equals 1”

25. Is there a pattern? 2 -1 = 1 2121 2 -2 = 1 2 1 2 X= 1 2 2 -3 = 1 2 1 2 1 2 XX = 1 2323 Keep going…. 1 4 1 8 = =

26. ExLaw #5 X -n = 1 XnXn “Eliminate any negative exponents by inverting the power”

27. For Example: Simplify (7) 0 =1 3 -2 = 1 3232 = 1 9

28. 8 -1 = 1 8181 1 8 = 1 = 8 -2 8282 1 =64

29. Page 9 [1-9] every other letter