1. Exponent
An exponent is a positive integer superscript written to the right of a number telling how many times that number is used as a factor.

2. Exponent The repeated factor is the base. 2 • 2 • 2 • 2 = 24 exponent
factors
base

3. Note:
2 = 21

4. A = s2 5
A = 52 5

5. V = s3 5
V = 53 5 5

6. Example 1
Write 6 × 6 × 6 × 6 using exponential notation. 64

7. Variable
A symbol used to represent an unknown number

8. Example 2
Write each of the following in expanded form. Simplify if possible.
a. 52 =
5 × 5 = 25

9. Example 2
Write each of the following in expanded form. Simplify if possible.
b. x6 =
x • x • x • x • x • x

10. Example 2
Write each of the following in expanded form. Simplify if possible.
−4(−4)(−4) = 16(−4) = −64
c. (−4)3 =

11. (−3)2 =
−3 • −3

12. (−3)2 =
−3 • −3
Note:
−32 is NOT the same as (−3)2.
−32 =
−(32) = −9

13. Example 3 Simplify each of the following. a. 34 − 42 =
(3 × 3 × 3 × 3) − (4 × 4)
= 81 − 16
= 65

14. Example 3 Simplify each of the following. b. −52 + (−2)2 =
−[5(5)] + (−2)(−2)
= −
= −21

15. Example 3 Simplify each of the following. c. 4(151)(23) =
4(15)(2 × 2 × 2)
= 60(8)
= 480

16. Multiplication Property of Exponents
For any number x and integers a and b, xa • xb = xa + b.

17. Example 4
Multiply 23 • 24. 27
23 • 24 = =

18. Power Property of Exponents
For any number x and integers a and b, (xa)b = xab.

19. Example
Simplify. Write as a standard numeral. 64
(22)3 =
26 =

20. Example
Simplify. Write as a standard numeral.
16x8
(2x2)4 =
24x8 =

21. Division Property of Exponents
For any number x and integers a and b, xa xb
= xa − b.

22. Example 5
Simplify the following. Leave answers in exponential form. a = 33
35 − 2 =

23. Example 5
Simplify the following. Leave answers in exponential form.
y7 y1 =
b. y7 y = y6
y7 − 1 =

24. =
34 34
3 • 3 • 3 • 3 = 1
3 • 3 • 3 • 3
So,
34 − 4 = 30 = 1

25. Properties of Exponents
xa • xb = xa + b
(xa)b = xab
xa ÷ xb = xa − b

26. Zero Power Any nonzero number to the zero power equals 1.
In symbols, n0 = 1 for any real number n, where n ≠ 0.
Note: 00 = undefined.

27. 42 45
4 • 4
1 43 = =
4 • 4 • 4 • 4 • 4
= 42 − 5 = 4−3
42 45

28. Negative Exponent
For any nonzero number x and any integer a, x−a = xa

29. 1 8 Example 6 Write the expression 2−3 without exponents and simplify.
1 23
1 2 • 2 • 2 = = 8 1
2−3 =

30. Example 7
Write the expression (−2)−5 without exponents and simplify.
1 (−2)5
(−2)−5 =
1 −2(−2)(−2)(−2)(−2) = = 32 -1

31. Example 8
Write the expression with a negative exponent and simplify.
1 x5 =
1 x5
x-5