Integer Exponents
As we have seen, a positive integer exponent indicates repeated
multiplication of a number.
For example, 45 indicates five factors, and each factor is 4.
• The base, 4, is the repeated factor.
• The exponent, 5, indicates the number of times the base appears as a
factor.
The base can also be a variable, as in x6.
The exponent can also be a variable, as in xn.
Example 1
For each of the following, identify the base and the exponent. Then simplify:
Solution
| a. The base is 3 and the exponent is 4.
34 indicates 4 factors. Each is 3. |
34 = 3 · 3 · 3 ·
3 = 81 |
b. The base is
 and the exponent is 2.
There are two factors. Each is
. |
 |
| c. The base is 2w and the exponent is 3.
There are three factors. Each is 2w.
|
(2w)3 = (2w) ·
(2w) · (2w)
= (2 · 2 · 2) · (w · w · w)
= 8w3 |
In 2w6, the base is w:
2w6 = 2 · w · w · w · w · w · w
In (2w)6, the base is 2w:
(2w)6 = 2w
· 2w · 2w · 2w · 2w · 2w |
Therefore,(2w)3 = 8w3.
d. Because there are no parentheses in the expression 2w6, the base is w,
not 2w. The exponent, 6, does not affect the 2. Thus, 2w6 cannot be
simplified.
| e. In -24, the base is 2, not -2. The
exponent is 4. Thus, there are four factors. Each is 2. |
-24
= -(2 · 2 · 2 · 2)
= -16 |
Here's a different example. In this case,
the base is negative two:
(-2)4 = (-2)(-2)(-2)(-2) = +16
|
Now, let’s look at exponents of 1 and 0, and negative exponents.
• If an exponent is 1, then the base can be written without an exponent. For example, 51 = 5 and x1 = x.
• If an exponent is 0, then the value of the expression is 1. For example, 50 = 1 and x0 = 1 (here, x
≠ 0).
• A negative exponent indicates a fraction. For example,
 . In general,
(here, x ≠ 0).
It is also true that
For example,
Example 2
Find: 2-1(3x)0y1
| Solution |
2-1(3x)0y1 |
| The factor 2-1 can be written as a fraction. The factor (3x)0 can be written as 1.
The factor y1 can be written as y.
Simplify.
|
 |
Thus, the result is
 .
|
