### 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!