1. Algebra 1
Section 1.7

2. Definition For x  N, bx is the product of x factors of b.
The repeated factor, b, is the base.
The exponent [x] indicates how many times b is used as a factor.
When numbers are written this way, they are said to be in exponential form or exponential notation.

3. Exponents
63 = 6 × 6 × 6
“6 to the third power”
“6 cubed” 6

4. Exponents
52 = 5 × 5
“5 to the second power”
“5 squared” 5

5. Example 1 a. 2 • 2 • 2 • 2 • 2 • 2 • 2 = b. -3 • (-3) • (-3) • (-3) =27
(-3)4

6. An Important Distinction
-34 = -1 • 34 =
(-3)4 = -3 • (-3) • (-3) • (-3) =
-81 81

7. Example 2 Evaluate: a. 93 = 9 • 9 • 9 =
b. (-7)4 = -7 • (-7) • (-7) • (-7) =
c. -42 = -(4 • 4) =
729
2401
-16

8. Properties of Exponents
52 • 53
(5 • 5)(5 • 5 • 5)
52+3 55
85 ÷ 82
8 • 8 • 8 • 8 • 8
8 • 8
85-2 83

9. Properties of Exponents
Product Property: To multiply like bases, add the exponents.
xa • xb = xa+b

10. Properties of Exponents
Quotient Property: To divide like bases, subtract the exponents. xa xb
= xa-b for x  0

11. Properties of Exponents
Power Property: To raise a power to a power, multiply the exponents.
(xa)b = xab

12. Example 3 Leave in exponential form: a. 52 • 510 = b. 32 • 34 • 39 =
c. 24 ÷ 2 =
d. 105 ÷ 103 =
e. (53)6 =
52+10 = 512
= 315
24-1 = 23
105-3 = 102
53(6) = 518

13. Properties of Exponents
The Quotient Property allows us to give meaning to negative exponents. x2 x3
= x2-3 = x-1 x2 x3 1 x1 =

14. Properties of Exponents
Any nonzero number to the zero power equals 1.
When x  0, x0 = 1

15. Properties of Exponents
Negative exponents:
x-a = 1 xa
= xa 1
x-a

16. Example 4 Evaluate. (-2)-4 = 1 = (-2)4 16 1 b. -3-2 = - = - 32 9 1 c.
= - 1 9 32
b = c. = 1
6-3
63 = 216

17. Example 5 Simplify, leaving each answer in positive exponential form:
b. 2-3 • 2-2 = 1 32 =
3-9+7 = 3-2
2-3+(-2) = 2-5 1 25 =

18. Example 5 Simplify, leaving each answer in positive exponential form:
d. (5-3)4 =
4-2-(-3) = 41
5-3(4) = 5-12
= 4 1
512 =

19. Homework: pp