© 2016 MaThCliX® MaTh Learning CenTer
Exponents Explained
1. Negative Whole Number Exponents:
If a number x is raised to a negative exponent then that exponent represents how many times 1
is being divided by the number x:
𝑥 −1 =
1
𝑥
𝑥 −2 =
1
1
= 2
𝑥∙𝑥
𝑥
𝑥 −3 =
1
1
= 3
𝑥∙𝑥∙𝑥 𝑥
Notice a pattern. To get from 𝑥 −2 to 𝑥 −3 you divide by 𝑥. To get from 𝑥 −3 to 𝑥 −2 you multiply by 𝑥.
Dividing by 𝑥 decreases the exponent by one and multiplying by 𝑥 increases the exponent by one. This
𝑥𝑛
𝑥𝑚
satisfies Equation 1:
= 𝑥 𝑛−𝑚 discussed in the previous worksheet on positive whole exponents.
𝑥 −2
= 𝑥 −2−1 = 𝑥−3
𝑥1
Notice that to reverse the process, the following equation applies:
Equation 2: 𝑥 𝑛 𝑥 𝑚 = 𝑥 𝑛+𝑚
𝑥 −3 ∙ 𝑥 1 = 𝑥 −3+1 = 𝑥 −2
2. Fractional Exponents:
The nth root of a number is equivalent to the number raised to a fractional exponent whose numerator
is 1 and denominator is equal to n:
1
𝑛
Equation 3: √𝑥 = 𝑥 𝑛
For example:
1
2
√𝑥 = √𝑥 = 𝑥 2
As you might know, the square root of a number x represents a quantity that is equal to the number x
when multiplied by itself.
𝑥 = √𝑥 ∙ √𝑥
We can prove this using equation 2:
1
1
1 1
+
√𝑥 ∙ √𝑥 = 𝑥 2 ∙ 𝑥 2 = 𝑥 2 2 = 𝑥 1 = 𝑥
© 2016 MaThCliX® MaTh Learning CenTer
This is also true for cube roots in respect that the cube root of a number x represents a quantity that is
equal to the number x when multiplied by itself three times. From equation 3 we see that n represents
1
𝑛
how many times the expression √𝑥 or 𝑥 𝑛 is multiplying one to equal x.
Now that we know what the denominator of a fractional exponent represents, let’s focus on the
numerator. Before we can do this, we have to talk about what happens when you have a base and its
exponent raised to another exponent. Equation 4 defines this expression where m and n are arbitrary
constants.
Equation 4: (𝒙𝒂 )𝒃 = 𝒙𝒂𝒃
𝑚
From equation 4, we see that the numerator of a fractional exponent such as m in 𝑥 𝑛 represents how
1
many times 𝑥 𝑛 is multiplying 1. In other words, if we factor m out, we can see that it is just an exponent
1
1
𝑚
of 𝑥 𝑛 . The same can be said about 𝑛 which can be written as 𝑛−1. If we factor this out of 𝑥 𝑛 , 𝑛−1 clearly
represents the nth root of 𝑥 𝑚 . Both of these examples are shown below.
𝑚
1
𝑛
𝑥 𝑛 = (𝑥 𝑚 )𝑛 = √𝑥 𝑚
𝑚
1
𝑥 𝑛 = (𝑥 𝑛 )𝑚
Both of these methods are crucial when evaluating numbers raised to fractional exponents as shown in
the preceding examples:
2
1
5
1
3
1. 83 = (83 )2 = ( √8)2 = (2)2 = 4
2
2. 92 = (92 )5 = ( √9)5 = 35 = 81 ∙ 3 = 243
So what about decimal exponents? Well, if you can convert the decimal into a fraction, for example .1
1
into 10 then you can use the method above. If not, then you better go find a calculator.