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
first or watch the
or try out the
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.
Basically, fractions are just numbers. The law of exponents that you have learned in
Exponent Laws-Part 1
Exponent Laws-Part 2
can still be used without any problems.
Now, watch the following math video to learn more.
Click play to watch video
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.
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.
on simplifying fractional exponents
on simplifying fractional exponents
Here are more lessons that you might be interested:
Exponent Laws - Part 1: First three laws
Exponent Laws - Part 2: Next three laws
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