The document reviews exponents and exponential notation. It defines the base as the starting number being used to an exponent, and the exponent as the number of times the base is used as a factor. It then provides rules for multiplying, dividing, and raising exponential expressions to a power: - When multiplying expressions with the same base, add the exponents - When dividing expressions with the same base, subtract the exponents - When raising an expression to a power, multiply the exponents It also notes that any number to the zero power equals one, and negative exponents represent the reciprocal of the base number rather than making the expression negative.

1. Exponents

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

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

4. Write 6 3 in standard form. 6 3 = 6  6  6 = 216

5. Write 6 4 in standard form. 6 4 = 6  6  6  6 = 1,296

6. What happens if we multiply 6 3  6 4 ? 6 3  6 4 = (6  6  6)  (6  6  6  6) 6 3 6 4

7. So… 6 3  6 4 = 6 7 because we used 6 as a factor seven times! Notice anything?

8. 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

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

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

11. Ex : 9 4  9 8 = ? 9 12

12. 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!

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

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

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

16. Ex: 6 8 ÷ 6 5 = ? 6 3

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

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

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

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

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

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

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

24. 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.

25. 5 -2 = = 2 -4 =