Integer Exponents
As we have seen, a positive integer exponent indicates repeated
multiplication of a number.
For example, 4^{5} 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
factor.
The base can also be a variable, as in x^{6}.
The exponent can also be a variable, as in x^{n}.
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.
3^{4} indicates 4 factors. Each is 3. 
3^{4} = 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)^{3} = (2w) Â·
(2w) Â· (2w)
= (2 Â· 2 Â· 2) Â· (w Â· w Â· w)
= 8w^{3} 
In 2w^{6}, the base is w:
2w^{6} = 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} = 8w^{3}.
d. Because there are no parentheses in the expression 2w^{6}, the base is w,
not 2w. The exponent, 6, does not affect the 2. Thus, 2w^{6} cannot be
simplified.
e. In 2^{4}, the base is 2, not 2. The
exponent is 4. Thus, there are four factors. Each is 2. 
2^{4}
= (2 Â· 2 Â· 2 Â· 2)
= 16 
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, 5^{1} = 5 and x^{1} = x.
â€¢ If an exponent is 0, then the value of the expression is 1. For example, 5^{0} = 1 and x^{0} = 1 (here, x
≠ 0).
â€¢ 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)^{0}y^{1}
Solution 
2^{1}(3x)^{0}y^{1} 
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
.
