Operations With Exponents
An exponent indicates how many times a number is used as a factor.
1
EXAMPLE
Write using exponents.
a. 3 ? 3 ? 3 ? 3 ? 3 = 35
b. a ? a ? b ? b ? b ? b = a 2b4
five factors of 3
2n ⴝ ■
two factors of a; four factors of b
The patterns shown at the right indicate that a 0 = 1 and that a -n = a1n .
2
EXAMPLE
Write each expression so that all exponents are positive.
3
a. a -2 b 3 = 12 ? b 3 = b2
a
10n ⴝ ■
22 ⫽ 4
102 ⫽ 100
21 ⫽ 2
101 ⫽ 10
20 ⫽ 1
100 ⫽ 1
2⫺1 ⫽ 12
1
10⫺1 ⫽ 10
2⫺2 ⫽ 14
1
10⫺2 ⫽ 100
3
b. x 3y 0z-1 = x 3 ? 1 ? 1z = xz
a
You can simplify expressions that contain powers with the same base.
3
EXAMPLE
Simplify each expression.
a. b 5 ? b 3 = b 5 + 3
= b8
Add exponents to multiply
powers with the same base.
5
b. x7 = x 5 - 7
x
= x -2 = 12
x
Subtract exponents to
divide powers with the
same base.
You can simplify expressions that contain parentheses and exponents.
4
EXAMPLE
Simplify each expression.
3
a 3b3 Raise each factor in the
a. Q ab
nR 5 3
n
parentheses to the third power.
b. A c 2 B 4 = c 2 ? 4 = c 8
Multiply exponents to raise a
power to a power.
EXERCISES
Write each expression using exponents.
1. x ? x ? x
2. x ? x ? x ? y ? y
3. a ? a ? a ? a ? b
4. a ? ba ?? ba ? a
7. x 5y-7z-3
8. ab-1c 2
Write each expression so that all exponents are positive.
5. c -4
6. m-2n0
Simplify each expression. Use positive exponents.
9. d 2d 6
10. n 4n
11. r 3 ? r 2 ? s7 ? s
12. x 5y 2 ? xy 2z6
5
13. a 2
7
14. cc
3
15. n6
5 3
16. a b8
17. (rt)3
18. (3x)2
21. A c 3 B 4
2 3
22. Q x5 R
4
19. Q ba R
20. Q y R
27. A 4a2b B 3(ab)2
28. A mg 3 B -1
a
25.
(2a4)(3a 2)
6a 3
29. g-3g-1
852
Skills Handbook
y
26. A x-2 B 3
30.
x 2y 3z21
x 5yz 3
n
23. A u4v2 B 3
(3a 3) 2
31. 18a
ab
xz 6
24. A p5 B -2
c 3d 7
32. 23
21
c
d