Fractional Exponent

Lesson Objective

This lesson shows you the basics behind fractional exponents and how they are related to roots. This lesson is divided into study tips, math video and practice questions.

About This Lesson

So far, we have been using integer exponents. Now, exponents can also be in the form of fractions (rational).

From this lesson you will realize that fractional exponents are closely related to square roots, cube roots and so on.

You can proceed by reading the study tips first or watch the math video or try out the practice questions.

$Formula relating fractional exponents with roots$

Study Tips

Tip #1

$Formula relating fractional exponents with roots$

It is important to understand the formula shown below before using it. This is because, you will be more comfortable applying the formula once you have understood it.

Tip #2

Basically, fractions are just numbers. The law of exponents that you have learned in Exponent Laws-Part 1 and Exponent Laws-Part 2 can still be used without any problems.

Now, watch the following math video to learn more.

Math Video

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Math Video Transcript

Fractional Exponents Transcript

00:00:01.220
So far, we have been dealing with integer exponents. 

00:00:05.170
This lesson will show you that exponents can be in term of fractions, and some of the basic ideas behind.

00:00:12.060
Let's start. We know that root 4 is equals to 2.

00:00:16.070
Now, fractional exponents are not very different from what you have seen here.

00:00:20.070
Let me show you why. Consider this term, 4 to the power of 1 over 2

00:00:26.090
We can rewrite 4 as 2 to the power of 2. So, now we have bracket 2 to the power of 2, to the power of 1 over 2.

00:00:36.080
Now, using the third exponent law, this term becomes, 2 to the power of 2 multiply by 1 over 2.

00:00:44.130
Now, both of the two cancel off,  and we are left with 2.

00:00:49.180
Notice that, these 2 numbers are the same.

00:00:53.200
Therefore, we can say that, root 4 is equals to 4 to the power of 1 over 2.

00:01:00.040
Most importantly, we can conclude that, the root sign is equivalent to this fractional exponent, 1 over 2.

00:01:07.160
Let's build up from here.  If we change 4 to 'a', root 'a' will be equals to 'a' to the power of 1 over 2.

00:01:16.130
If the root is changed to cube root, the exponent will change to, 1 over 3. Do you see the pattern here?

00:01:22.210
Finally, if we change 3 to m, the exponent will change to 1 over m.

00:01:29.010
So, in general, the m root of 'a' is equals to 'a' to the power of 1 over 'm'.

00:01:35.150
Let's analyze this formula further. If 'a' is changed to 'a' to the power of n, we will get bracket 'a' to the power of 1 over m.

00:01:46.060
Again, using this exponent law,  this term becomes 'a' to the power of 'n' multiply by 1 over 'm'. Hence, we now have 'A' to the power of n over m.

00:01:58.180
From here, we can see that, 'a' to the power of n over m is equals to, 'm' root of 'a' to the power of n. 

00:02:07.100
Let's rewrite these terms here.

00:02:11.210
Let's continue, notice that we can switch the two terms here to get, 'A' to the power of 1 over m multiply by n.

00:02:20.120
Now, notice that we have 'a' to the power of 1 over m. If we look carefully, this part of the term is also equals to m root of 'a'.

00:02:32.120
Therefore, this term can be written as, bracket m root to the power of n.

00:02:39.190
With this, we can see that, bracket m root 'a' to the power n is equals to, m root a to the power of n.

00:02:48.070
Let's write down this observation here.

00:02:51.160
Finally, we have the formula, 'a' to the power of n over m, equals to m root 'a' to the power of n, equals to bracket m root 'a' to the power of n.

00:03:03.030
Now, to understand this formula better, let's simplify, 8 to the power of 2 over 3.

00:03:10.060

Using this formula, 'a' to the power of n over m, equals to bracket m root n to the power of n, we get 8 to the power of 2 over 3 as, bracket 3 root 8 to the power of 2. 

00:03:23.070
Root 3 of 8 is 2. Finally, bracket 2 to power of 2 gives 4.

00:03:30.210
Now, without using this formula, we can also simplify this example using the usual exponent laws. Let me show you how.

00:03:40.100
We know that 8 is equals to, 2 to the power of 3. So let's replace 8 with 2 to the power of 3.

00:03:49.060
Let's use the third exponent law to further simplify this term. Bracket 2 to the power of 3, to the power of 2 over 3, gives 2 to the power of 3 multiply by 2 over 3.

00:04:01.210
When we multiply 3 with 2 over 3, both threes cancels off. This leaves us with  2 to the power of 2.

00:04:09.140
Now, 2 to the power of 2 gives 4. This is the same answer as the previous method.

00:04:15.160
Personally, I think this way of simplifying is more elegant. But, it's up to you to choose the way that suits you.

00:04:22.190
That's all for this lesson on fractional exponents. Try out the practice question to further your understanding.

Practice Questions & More

Multiple Choice Questions (MCQ)

Now, let's try some MCQ questions to understand this lesson better.

You can start by going through the series of questions on Fractional Exponent or pick your choice of question below.

Question 1 on simplifying fractional exponents
Question 2 on simplifying fractional exponents

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