1. Section 5.1
The Rules of Exponents

2. Exponents
An exponent is a “shorthand” number that saves writing the multiplication of the same numbers.
exponent 34
base
This is read “three to the fourth power.” 2

3. The Product Rule The Product Rule
To multiply two exponential expressions that have the same base, keep the base and add the exponents.
xa · xb = xa + b 3

4. Example Multiply. Numerical coefficient a. c4 · c5 = c4 + 5 = c9
b. 3a3 · a6
= 3a3 + 6 = 3a9
c. 4w2 · 2w5
= (4)(2)w2 + 5 = 8w7 4

5. Example
Multiply.

6. The Quotient Rule The Quotient Rule (Part 1)
Use this form if the larger exponent is in the numerator and x  0. 6

7. Example
Divide. a.
Note that the base does not change. b. 7

8. The Quotient Rule The Quotient Rule (Part 2)
Use this form if the larger exponent is in the denominator and x  0. 8

9. Example
Divide. a.
Note that the base does not change. b. 9

10. Example
Divide. a. b.

11. The Quotient Rule The Quotient Rule (Part 3)
if x  0 (00 remains undefined). 11

12. Example
Divide. a. b. 12

13. Raising a Power Raising a Power to a Power
To raise a power to a power, keep the same base and multiply the exponents. 13

14. Example
Simplify.
a. (x5)3
= x5·3 = x15
b. (y3)3
= y3·3 = y9 14

15. Raising a Power
Product Raised to a Power 15

16. Example Simplify. a. (2c)3 = (2)3c3 = 8c3 b. (5xy)2
= (5)2(xy)2 = 25x2y2
c. (4x3y2)3
= 43x9y6 = 64x9y6 16

17. Raising a Power
Quotient Raised to a Power
if y ≠ 0. 17

18. Example
Simplify. a. b. 18

19. Example
Simplify.
Simplify inside the parenthesis first. Note that z0 = 1.
Simplify the coefficient.