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.
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.
Now, watch the following math video to learn more.
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Math Video Transcript
Fractional Exponents Transcript
So far, we have been dealing with integer exponents.
This lesson will show you that exponents can be in term of fractions, and some of the basic ideas behind.
Let's start. We know that root 4 is equals to 2.
Now, fractional exponents are not very different from what you have seen here.
Let me show you why. Consider this term, 4 to the power of 1 over 2
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.
Now, using the third exponent law, this term becomes, 2 to the power of 2 multiply by 1 over 2.
Now, both of the two cancel off, and we are left with 2.
Notice that, these 2 numbers are the same.
Therefore, we can say that, root 4 is equals to 4 to the power of 1 over 2.
Most importantly, we can conclude that, the root sign is equivalent to this fractional exponent, 1 over 2.
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.
If the root is changed to cube root, the exponent will change to, 1 over 3. Do you see the pattern here?
Finally, if we change 3 to m, the exponent will change to 1 over m.
So, in general, the m root of 'a' is equals to 'a' to the power of 1 over 'm'.
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.
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.
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.
Let's rewrite these terms here.
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.
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'.
Therefore, this term can be written as, bracket m root to the power of n.
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.
Let's write down this observation here.
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.
Now, to understand this formula better, let's simplify, 8 to the power of 2 over 3.
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.
Root 3 of 8 is 2. Finally, bracket 2 to power of 2 gives 4.
Now, without using this formula, we can also simplify this example using the usual exponent laws. Let me show you how.
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.
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.
When we multiply 3 with 2 over 3, both threes cancels off. This leaves us with 2 to the power of 2.
Now, 2 to the power of 2 gives 4. This is the same answer as the previous method.
Personally, I think this way of simplifying is more elegant. But, it's up to you to choose the way that suits you.
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.
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