1. Warm-up, 3/28 Compute: 1. 2 x 2 x 2 x 2 = 2. 3 x 3 x 3 = 3. 2 x 2 x 3 x 3 x 3 = 4. 5 x 5 x 2 x 2 = 5. 2 x 2 x 4 =

2. Exponents

3. Review: What do exponents tell us? What does 5 3 mean? 5 3 =5  5  5

4. What is the “base”? What is the “exponent”? 8 58 5 Base Exponent

5. Write 6 3 in standard form. 4 3 = 4  4  4 = 64

6. Write 6 4 in standard form. 3 4 = 3  3  3  3 = 81

7. What happens if we multiply 2 3  2 4 ? 2 3  2 4 = (2  2  2)  (2  2  2  2) 2323 2424

8. So… 2 3  2 4 = 2 7 because we used 2 as a factor seven times! Notice anything?

9. Let’s try another example: What is 5 2  5 4 ? 5 2  5 4 = (5  5)  (5  5  5  5) or 5 2  5 4 = 5 6

10. For MULTIPLICATION: x s  x t = x s + t (For any number x and for integers s and t.)

11. In other words, when multiplying two numbers in exponential form with the same base, add the exponents.

12. Ex: 9 4  9 8 = ? 9 12

13. What happens if we divide? 7 5 7 3 7 5 = 7  7  7  7  7 7 3 7  7  7 Cancel out any numbers that are in the numerator AND denominator!

14. After we cancel, we are left with 7 2. So, 7 5 = 7 2 7 3

15. For DIVISION: x s = x t (For any nonzero number x and for integers s and t.) x s - t

16. In other words, when dividing two numbers in exponential form with the same base, subtract the exponents.

17. Ex: 6 8 ÷ 6 5 = ? 6 3

18. What about (5 2 ) 3 ? (5 2 ) 3 (5 2 )  (5 2 )  (5 2 ) (5  5)  (5  5)  (5  5) 5 6

19. Raising a power to a power: (x s ) t = x s  t (For any number x and for integers s and t.)

20. In other words, when raising a number with an exponent to a power, multiply the exponents.

21. Ex: (5 3 ) 7 = ? 5 21

22. One important thing to remember… x 0 = 1 (Any number to the zero power is equal to 1!)

23. Why??? For example… 3 2 ÷ 3 2 = 3 2 - 2 = 3 0 3 2 = 3  3 = 9 = 1 3 2 3  3 9

24. So, 3 0 = 1 This works for any base number!

25. Can we have negative exponents? Yes! Negative exponents represent the reciprocal. The reciprocal of x is The reciprocal of 10 is They do NOT make the answer negative.

26. 5 -2 = = 2 -4 =

27. Review: Compute: 2 5 Answer: 32

28. Compute: 5 -2 Answer: 1/25

29. Compute: 0 2 Answer: 0

30. Compute: 2 0 Answer: 1

31. Compute: - 4 2 Answer: -16

32. Compute: (-4) 2 Answer: 16

33. Compute: - 4 -2 Answer: - 1/16

34. Compute: 3 -2 Answer: 1/9

35. Compute: (2 2 )(2 3 ) Answer: 2 5 or 32

36. Compute: (3 3 ) 2 Answer: 3 6 or 729

37. Multiply and express in exponential form: (4 5 )(8 2 ) Answer: 2 16

38. What is one-third of 3 99 ? Answer: 3 98