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Video: Fractional Powers

Tim Burnham

Here, we introduce and explain the derivation and usage of fractional powers, or rational exponents, through a series of increasingly complex examples.

14:38

Video Transcript

In this video, we’re going to look at fractional powers or sometimes they’re called “rational exponents” depending on where you live. So this is situations where we’ve got something to the power of something else; that something else is a fraction or a rational expression. So one over 𝑎 or 𝑎 over 𝑏 or a third or two-thirds or three-quarters, and so on. Firstly, you should already know about powers or indices or exponents’ notation. We have a base. And the superscript number tells us how many times to write a base number down and multiply them together. So three squared means three times three; write down three twice and multiply them together. Three cubed means three times three times three; write the three down three times and multiply them together.

You should also remember the addition rule of powers. So if we have three to the power of four times three to the power of five, that means three times three times three times three times three times three times three times three times three. So we have four threes multiplied together here and five threes multiplied together here. So we can write that as three to the power of four plus five adding the powers, which gives us three to the power of nine altogether. So we can sum this up by saying when we multiply numbers in power form with the same base, we can just add the powers together. One more quick example just to make sure you’ve remembered all that: if we had five to the power of seven times five to the power of twelve, we just add the seven and twelve; we got nineteen fives multiplied together in total.

And now onto the main point of this video — fractional powers or rational exponents. Right, let’s start by looking at a little example. Evaluate nine to the power of a half times nine to the power of a half. Well, when multiplying numbers with powers in the same base, we just need to add the powers. So nine to the power of a half times nine to the power of a half is equal to? So nine to the power of a half times times [nine] to the power of a half is nine to the power of a half plus a half, which is nine to the power of one which of course is just nine. So we’ve got something, nine to the power of a half, times itself, nine to the power of a half and that gives us an answer of nine. Well, what is it that when we times it by itself we get an answer of nine? Clearly that’s three; so nine to the power of a half must be three. Now one thing to just be a bit aware of, a lot of people see nine to the power of a half and think it means nine times a half, which would be four and a half. But we’ve just seen that isn’t true. So that’s probably the biggest mistake that people make here; so be careful not to make that mistake.

So if we look back at our example, what we’re describing here three times three is equal to nine. We’re really describing square roots. What is it that when we times it by itself we get another number? We’re talking about square roots; so nine to the power of a half which is three is equal to the square root of nine. That’s the definition: when we got something to the power of a half, it means the square root of that number. So 𝑥 to the power of a half, a number to the power of a half, means the square root of 𝑥, the square root of that number. That’s the main learning point of this part of the video. Now that’s because when I multiply 𝑥 to the power of a half by itself, what I end up doing is adding those powers together and getting 𝑥 to the power of one, which is just 𝑥. So when I multiply something by itself, I get 𝑥; that must mean that this thing is the square root of 𝑥.

Okay, let’s move on to this example. Evaluate twenty-seven to the power of a third times twenty-seven to the power of a third times twenty-seven to the power of a third. Well, this is just twenty-seven to the power of a third plus a third plus a third, which is twenty-seven to the power of one which is just twenty-seven. So this means that something times itself times itself again gives us an answer of twenty-seven. And this again is-is three in fact; three times three is nine times three is twenty-seven. So twenty-seven to the power of a third must be equal to three. And what we’ve done is we’ve shown that twenty-seven to the power of a third means the cube root of twenty-seven. What is it that when I multiply it by itself and then by itself again and I get twenty-seven? That’s the definition of cube roots. So 𝑥 to the power of a third means the cube root of 𝑥 because 𝑥 to the power of a third times itself times itself again means 𝑥 to the power of a third plus a third plus a third, which is 𝑥 to the power of one which is just 𝑥. That’s the definition of cube roots. Now in general that means that 𝑥 to the power of one over 𝑎 is the 𝑎th root of 𝑥. So if I had five to the power of one over seven, we’d be looking for the seventh root of five.

And another example, thirty-two to the power of a fifth is equal to the fifth root of thirty-two, which is equal to two because two times two times two times two times two is equal to thirty-two. Now that’s because when I multiply thirty-two to the power of a fifth times thirty-two to the power of a fifth times thirty-two to the power of a fifth times thirty-two to the power of a fifth times thirty-two to the power of a fifth, I just have to add the powers together. A fifth plus a fifth plus a fifth plus a fifth plus a fifth is just one; so that’s thirty-two to the power of one or just thirty-two. And what is it that when I multiplied by itself and by itself again by itself again by itself again I get an answer of thirty-two? Well it’s the fifth root of thirty-two. And as we said before that is in fact two because two times two times two times two times two is equal to thirty-two.

Okay, hopefully that’s okay. Now let’s look at examples like this: eight to the power of two over three. We’re gonna look at what that means. Well we’ve just seen that the number on the bottom of the fraction of the power tells us which root it is; so we’ve got a three. So a three on the denominator here tells us that it’s a cube root and the two on the numerator of the power tells us that we’re gonna square it. If that was a five on the top of that fraction there, then it would be to the power of five. So one other way of writing that: we’re looking at the cube root; so that’s the cube root of eight and then we’re squaring that answer; so it’s the cube root of eight all squared. In fact, we’d get the same answer if we did this over here — if we took eight squared and then took the cube root of that. But the first version of those is usually the easiest one to work with: we take the cube root of eight first, we get a smaller number, and then we square that; that’s easier than trying to find the cube root of a bigger number, eight squared.

But let’s just check that. So in this case, the cube root of eight is two; so this here is two squared, which is equal to four. And in the other case, the cube root of eight squared, eight squared is sixty-four. And we’re looking for the cube root of sixty-four, so four times four times four is sixty-four. So that would also give us an answer of four. I’m just to show you that you’ll always get the same answer no matter which approach you’ll use. So let’s just look at that in a slightly different way: eight to the power of two-thirds. Well that’s the same as eight to the power of one-third plus one-third. So one-third plus one-third is clearly two-thirds. And we know from our addition rule that eight to the power of a third plus a third is the same as eight to the power of a third times eight to the power of a third. Well that’s eight to the power of a third all squared times itself, and we said that eight to the power of a third is the cube root of eight. So hopefully that all makes logical sense.

Okay, let’s just try a couple of examples and see it in action. So evaluate a hundred to the power of five over two. Well, this is gonna be the square root of a hundred all to the power of five because we had a two on the denominator and a five on the numerator. So the square root of a hundred is ten and ten to the power of five is one hundred thousand; so there’s our answer. So the next one is twenty-seven to the power of four over three. So we’ve got a fractional power four on the top means it’s gonna be to the power of four. Three on the bottom means we’re gonna take the cube root; so that’s the cube root of twenty-seven all to the power of four. And the cube root of twenty-seven is just three; so this is three to the power of four. And three to the power of four means three times three times three times three. So three times three is nine, three times three is nine, and nine nines are eighty-one.

Okay, let’s shake it up a bit then and hopefully scare you a little bit. So let’s try it with a fractional base as well as a fractional power. Eight over twenty-seven to the power of two over three. Now we don’t need to panic because when we’ve got fractions to powers, it’s just the numerator to that power over the denominator to that power. So there’s that first step and then the three on the denominator of the power means it’s the cube root and the two on the numerator of the power means that it’s squared. So on the top here, we’ve got the cube root of eight all squared and on the bottom here, we’ve got the cube root of twenty-seven all squared. Now I really would strongly recommend actually writing all these stages out as you go, especially in a test or an exam; otherwise, it’s so easy to lose track of what these numbers mean. Now the cube root of eight is two; so that becomes two squared on the top and the cube root of twenty-seven is three; so that becomes three squared on the bottom. And two squared is four; three squared is nine. So our answer is four over nine.

Right, let’s have one last example then. We’re gonna throw decimals into the mix as well now. Decimals can be represented as fractions. And I would strongly recommend that that’s how we approach this one: nought point one two five is one over eight. So we can rewrite this question like this; so that’s one over eight to the power of five over three. And again we can split up the numerator and the denominator; so that’s one to the power of five over three on the top and eight to the power of five over three on the bottom. So the three on the denominator of the power means the cube root and the five means to the power of five; so we’ve got the cube root of one all to the power of five over the cube root of eight all to the power of five. Now it doesn’t matter how many times I divide one by itself or multiply it by itself; I’m always gonna get an answer of one. So I know that the numerator there is one; the cube root of eight is two. And we’ve got to do two to the power of five, and when I do two times two times two times two times two, I get an answer of thirty-two; so this is one over thirty-two.

So we’ve introduced the idea fractional powers. And we sense some really quite tricky questions; we sense a quite basic questions as well. But we’ve also included fractional bases and decimal bases, but the basic rule is just like this 𝑥 to the power of one over 𝑎 is equal to the 𝑎th root of 𝑥. So for example, nine to the power of a half is the square root of nine. We’ve got two on the denominator of the power; so it’s the square root of nine, and the square root of nine is three. And if we’ve got fractional pairs that look like this, 𝑥 to the power of 𝑎 over 𝑏, so the numerator of the power isn’t one. That means the 𝑏th root of 𝑥 all to the power of 𝑎 or we could do 𝑥 to the power of 𝑎 first and take the 𝑏th root of that.

The issue here is that this version is usually easier to work with because you’re ending up with smaller numbers to work with in the first place. So for example, sixteen to the power of three quarters means the fourth root of sixteen because the four is on the denominator of the power and then we take that answer and cube it. That’s two to the power of three and two to the power of three is eight. Just quickly to show you that the other way of doing that would have been the fourth root of sixteen cubed and sixteen cubed is four thousand and ninety-six. We’d be looking for the fourth root of four thousand and ninety-six. Now I’ll leave you to do a bit of work on your calculator just to check that this is in fact still eight.

And finally, if we have a fraction to the power of another fraction, we’ve got 𝑥 over 𝑦 all to the power of 𝑎 over 𝑏. So what I strongly recommend is splitting up the fraction into the top half and the bottom half. So this is the numerator 𝑥 to the 𝑎 over 𝑏 and we’ve got 𝑦 to the 𝑎 over 𝑏. Now 𝑥 to the 𝑎 over 𝑏 is the 𝑏th root of 𝑥 all to the power of 𝑎 and 𝑦 to the 𝑎 over 𝑏 is the 𝑏th root of 𝑦 all to the power of 𝑎. We can use this other format over here again, but they’re gonna probably give us harder numbers to work with. So I would definitely recommend using the first rather than the second version of that.

And just a quick example of that one, if we had four over nine to the power of three over two, that is the square root of four because we’ve got a two on the denominator of the power and that is cubed because we’ve got a three on the numerator of the power; so that’s the top number here. And then on the bottom of that fraction, we’ve got the nine. We’re taking the square root of nine and we’re cubing that answer. And the square root of four is two; so that top number becomes two cubed. And the square root of nine is three; so the bottom number becomes three cubed. And two cubed is eight and three cubed is twenty-seven. So that’s pretty much fractional powers. Time for you to go and do some exercises for yourself now. Good luck!