### 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!