Fractional powers

April 25, 2022

In this post

Rules of fractional powers

Instead of always having a whole number that is a power we may sometimes have one that is a fraction – hence the name fractional power. When this happens we must follow certain rules to work out answers. The same rules for powers do still apply, and if you wish to recap these they can be found in our earlier posts.

The main rule for when we have a fractional power is the following:

So when we have a power that has a number on the bottom of a fraction and a 1 as the numerator we can convert it into an equation that finds the nth root of the number a. This is the same as asking what number multiplied by itself n times will equal a. Please note that if there is no number above the root sign then a 2 is implied, so $\sqrt{a}$ is equal to $\sqrt[2]{a}$ .

Fractional Index Form	Root Form	Answer
$25^{\frac{1}{2}}$	$\sqrt{25}$	5
$64^{\frac{1}{3}}$	$\sqrt[3]{64}$	4
$81^\frac{1}{4}$	$\sqrt[4]{81}$	3
$64^{\frac{1}{6}}$	$\sqrt[6]{64}$	2

Now that we know how to find an answer when an indices has the form $\frac{1}{n}$ we can move on to looking at how an answer is found when the numerator is not 1. So how would we go about finding something that is to the power of $\frac{a}{b}$ ?

This is done by making use of a technique that we learnt all the way back at the start of this course. The fact that we can multiply two indices will help us to work out this sort of problem. We have already seen that $(a^n)^k=a^{n \times k}$ and we can use this information to do the opposite. This means that $a^{n \times k}$ can be separated and we will see that this is the same as .

By making use of this technique we can see that $n^{\frac{a}{b}}$ is really the same as having $(n^{\frac{1}{b}})^a$

We can then use this to find an answer for $n^{\frac{1}{b}}$ from the technique that we saw on the previous page and simply put the answer to the power of the unknown a.

Example

What is $125^{\frac{2}{3}}$ ?

By separating the power into a fraction with 1 as the numerator and a power on the outside we get $(125^{\frac{1}{3}})^2$ and we can now work out the value inside the bracket using the techniques shown earlier in the lesson:

(125^{\frac{1}{3}})^2=(\sqrt[3]{125})^2=(5)^2=25

By saving the power on the outside of the bracket until last we have made use of the BIDMAS rule.

Negative fractional indices

You will hopefully remember that when a whole number is put to the power of a negative we must find the ‘reciprocal’. This means that something like $9^{-2}$ is exactly the same as finding $\frac{1}{9^2}$ . We must simply put the entire calculation to the bottom of a fraction that has 1 as the numerator.

The same thing is done even if the power is a fraction. So for something like we can simply write as $\frac{1}{16^{\frac{3}{4}}}$

We can then work out the denominator of this fraction in the same way as in the previous example: by separating the power and calculating the answer. The only difference which a negative has in this type of problem is that it will put the answer on the bottom of a fraction with 1 at the top (it will give the reciprocal of the answer without a negative sign).

So the answer to $16^{-\frac{3}{4}}$ is the exact same as finding the reciprocal of $16^{\frac{3}{4}}$ . Using the explained techniques we would get the answer to $16^{\frac{3}{4}}$ as 8 so the answer to $16^{-\frac{3}{4}}$ would simply be the reciprocal of this, $\frac{1}{8}$ .

Fractional powers of a fraction

The same skills which we have already looked at here can be extended to be used when we have a fraction that is to the power of another fraction. Again, we must make use of some techniques learnt in module 1: the fact that a fraction to a power can be separated and both the numerator and denominator can be put to the power separately. This tells us that $(\frac{3}{4})^5$ is the same as having $\frac{3^5}{4^5}$ . And the same can be done for any different numbers in the example.

If the power was a fraction in a case like the above, then we would simply have one number to a fractional power that is divided by another. This would be no problem to solve as we could find the numerator and denominator separately using the techniques outlined in this lesson and then put the correct numbers into the fraction.

Example

Find the value of $(\frac{16}{81})^{\frac{3}{4}}$

Here we can multiply the top and bottom of the fraction by the power separately, and then work out the numerator and denominator as separate problems using the skills we have just learnt.

(\frac{16}{81})^{\frac{3}{4}}=\frac{16^{\frac{3}{4}}}{81^{\frac{3}{4}}}=\frac{(\sqrt[4]{16})^3}{(\sqrt[4]{81})^3}=\frac{2^3}{3^3}=\frac{8}{27}

By splitting the question into smaller parts and using different techniques on each we will eventually get to the final answer.

If the example above had a minus fractional power we would work in the same way except we would then find the reciprocal of the answer at the end. So .

(\frac{16}{82})^{-\frac{3}{4}}=(\frac{8}{27})^{-1}=\frac{27}{8}

Another thing that should not be forgotten is the fact that a fraction can be changed into a mixed number and vice versa. This then means that if we were to come across a question with a mixed number as the power we can easily change this to being a fraction and then work through the problem in the same way as we have seen in the previous examples.

So if you are faced with a question like $4^{1\frac{1}{3}}$ this can easily be changed to $4^{\frac{4}{3}}$ and can then be worked out using the same methods that we have outlined in this lesson.

Fractional powers

In this post

Rules of fractional powers

Example

Negative fractional indices

Fractional powers of a fraction

Example

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