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.
â€¢ The base, 4, is the repeated factor.
â€¢ The exponent, 5, indicates the number of times the base appears as a
The base can also be a variable, as in x6.
The exponent can also be a variable, as in xn.
For each of the following, identify the base and the exponent. Then simplify:
|a. The base is 3 and the exponent is 4.
34 indicates 4 factors. Each is 3.
= 3 Â· 3 Â· 3 Â·
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) Â·
(2w) Â· (2w)
= (2 Â· 2 Â· 2) Â· (w Â· w Â· w)
|In 2w6, the base is w:
2w6 = 2 Â· w Â· w Â· w Â· w Â· w Â· w
In (2w)6, the base is 2w:
(2w)6 = 2w
Â· 2w Â· 2w Â· 2w Â· 2w Â· 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
|e. In -24, the base is 2, not -2. The
exponent is 4. Thus, there are four factors. Each is 2.
= -(2 Â· 2 Â· 2 Â· 2)
|Here's a different example. In this case,
the base is negative two:
(-2)4 = (-2)(-2)(-2)(-2) = +16
Now, letâ€™s look at exponents of 1 and 0, and negative exponents.
â€¢ If an exponent is 1, then the base can be written without an exponent. For example, 51 = 5 and x1 = x.
â€¢ If an exponent is 0, then the value of the expression is 1. For example, 50 = 1 and x0 = 1 (here, x
â€¢ A negative exponent indicates a fraction. For example,
. In general,
(here, x ≠ 0).
It is also true that
|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.
Thus, the result is