© 2016 MaThCliX® MaTh Learning CenTer
Exponents Explained
1. Positive Whole Number Exponents:
First:
An exponent is a number placed as a superscript of a base which can be a number or variable.
When the exponent is a positive integer, it represents how many base terms are being
multiplied together.
𝑏𝑎𝑠𝑒 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡 = 𝑎𝑛𝑠𝑤𝑒𝑟
One base term: 𝑥 1 = 𝑥
Two base terms: 𝑥 2 = 𝑥 ∙ 𝑥
Three base terms: 𝑥 3 = 𝑥 ∙ 𝑥 ∙ 𝑥
51 = 5
52 = 5 ∙ 5 = 25
53 = 5 ∙ 5 ∙ 5 = 125
When the exponent is a positive whole number the process is straight forward, as shown
above. However, what if a number has an exponent that is equal to zero?
Well, if we look at the expressions above, we can see a pattern. To get from 𝑥 2 to 𝑥 3 we simply
multiply by 𝑥. Multiplying by x increases the exponent by 1. Going backwards, we can also conclude that
dividing by x decreases the exponent by 1.
Here is an example with numbers:
53
125
=
= 25 = 52
5
5
Furthermore, to get from 𝑥 3 to 𝑥 we divide by 𝑥 2 .
53 5 ∙ 5 ∙ 5
125
=
=
= 5(3−2) = 51 = 5
52
5∙5
25
Thus, the correct method when dividing x to any power n by x to any power m is to subtract the
exponent of x in the denominator from the exponent of x in the numerator. Equation 1 displays this:
𝑥𝑛
Equation 1: 𝑥 𝑚 = 𝑥 𝑛−𝑚
2. Zero As an Exponent
Using the above expression, if we have a number or variable to the power of zero, then solving
algebraically we get the following:
© 2016 MaThCliX® MaTh Learning CenTer
𝑥 0 = 𝑥 𝑛−𝑚
0=𝑛−𝑚
𝑚=𝑛
𝑥𝑛
If n and m are equal then the expression 𝑥 𝑚 is equal to 1 because the numerator and denominator are
equal.
Now, the expression 𝑥 0 = 1 is true but only when x does not equal 0. This is because we encounter a
huge problem when we try to solve 00 . This can be proved algebraically:
For 𝑥 𝑛−𝑚 =
𝑥𝑛
𝑥𝑚
, let’s assume that x is equal to zero and that n and m are equal and are whole numbers.
Then, 00 = 0𝑛−𝑚 =
0𝑛
0𝑚
0
= 0 = undefined
Anything divided by zero is an indeterminate expression.