# Lesson video

In progress...

Hi, my name is Mr. Chan.

And in this lesson, we're going to learn about fractional indices.

Let's begin by looking at an example where we should know how to simplify this expression here.

So we're going to use the multiplication law for indices to simplify this, 49 to the power of 1/2 multiplied by 49 to the power of 1/2.

Now I can see both base numbers are 49, the index numbers are 1/2, so we would add the index numbers together so we would get 49 to the power of 1/2 add 1/2, which would equal 49 to the power of one.

But I know 49 to the power of one is just simply 49.

Now what we can say with what we've started with is we've multiplied 49 to the power of 1/2 by itself.

So that's equivalent to squaring it.

And we found that it equals 49.

So to find out what 49 to the power of 1/2 is we can square root both sides of that little equation there.

So 49 to the power of 1/2 would equal the square root of 49, which equals seven.

Let's have a look at another example.

Generalising, we've got b the power of 1/2 multiplied by b to the power of 1/2.

So again, we're going to add the two index numbers together.

So that equals b the power of one.

So what I know now is b the power of 1/2 multiplied by b to the power of 1/2 is just simply b to the power of one, which is just b.

So trying to unravel what we've just discovered there, we've multiplied b to the power of 1/2 by itself, in other words, squaring it, and it equals b.

So from this little equation, we can find b the power of 1/2 and what that equals by square rooting both sides of the equation that equals the square root of b.

So what have you noticed? So let's look at another example, where we're going to use the multiplication law for indices.

Again we've got b to the power of 1/3 multiplied by b to the power of 1/3, multiplied by b to the power of 1/3.

Now, because all base numbers are the same, we would add the powers together.

So that would equal b to the power of 1/3, add 1/3, add 1/3.

And if we add those fractions together, we would get b to the power of one, which is just simply b.

Now what we've done there is we've got b to the power of 1/3 repeatedly multiplied by itself three times.

And that's the same as cubing.

So we've got b to the power of 1/3 cubed.

And we've found that equal to b.

So we can cube root both sides to actually find what b to the power 1/3 actually equals.

That would be the cube root of b.

So what you notice here, is that it doesn't really matter what base number we started with 'cause the base number remains constant throughout this example.

We could have used the base number five and we would have got the same amount.

So we could have used the base number of 10 and it would still equal the base number.

So let's try and apply that to actually working something useful out here.

We've got to calculate eight to the power of 1/3.

So the example we've just covered tells us that eight to the power of 1/3 would equal the cube root of eight.

Well, the cubed root of eight, we're looking for three numbers that multiply together that are the same to equal eight, that would equal two.

So what I know now is eight to the power of 1/3 equals two.

So the examples we've covered so far looked at a fractional index number of 1/2, and then a fractional index number of a 1/3.

How would you work out a number raised of the power of 1/4? So reviewing what we've looked at so far, b to the power of 1/2 would equal the square root of b, b to the power of 1/3 would equal the cube root of b, and how would we explain b to the power of 1/4? Would it equal the fourth root of b? Well, yes it would because let's look through this.

We can show that b to the power of 1/4 by repeatedly multiplying b to the power of 1/4 by itself four times.

That would indicate that we add all those four quarters together to equal b to the power of one, which is just b by itself.

So like I've said, we've repeatedly multiplied b to the power of 1/4 four times, that's the same as raising b to the power of 1/4 to the power of four.

So we would fall through both sides, and we can say that b to the power of 1/4 equals the fourth root of b.

Here's a question for you to try.

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So the examples covered answering all of these questions.

Remember that the power of 1/2 indicates that you're trying to find the square root, a number to the power of 1/3, you're trying to find the cube root, a number to the power of 1/4, you're trying to find the fourth root of that number.

Here's some more questions for you to try.

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If we look at part A as an example here, we can see that we're just applying what we've already learned about fractional indices to answering the question where we're adding two numbers together.

So nine to the power of 1/2 would be the square root of nine, so that's three.

We're adding on 81 to the power of 1/2, which is the square root of 81, which equals nine.

So really, this question is asking us what is three add nine? And that equals 12.

So let's try and review what we've learned so far.

We know that b to the power of 1/2 equals the square root of b.

b to the power of 1/3 equals the third root of b.

b to the power of 1/4 is the forth root of b.

What you should notice is that the index number, the denomination in the index number, tells you what we root the b by.

So if we have a denominator of two, for example, in b to the power of 1/2, we'd have the square root of b.

In b to the power of 1/3, the denominator in that fraction is three so we look for the third root of b, or the cube root of b.

And if we have b to the power of 1/4, the denominator is four, so that would equal the fourth root of b.

So if we generalise that to b to the power of 1/n, think about we're multiplying b to the power of n and raising that to the power of n, that would equal b.

So b to the power of 1/n would simply be the nth root of b.

So let's apply that in another example.

So we're going to try and apply what we've just covered in an example.

We've got to try and write the 12th root of p in index form.

So what I know about the 12th root of p is that that equals p.

Well, if I'm looking for the 12th root, that would mean that my denominator as a fractional index, would be 12.

So that would equal p to the power of 1/12.

Here's some questions for you to try.

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Remember that the denominator in the fraction tells us what root we're looking for.

So if we look at questions c, for example, we're looking for what power is the seventh root of c, because the denominator would tell us that it must be seven, the answer would be c to the power 1/7.

Here's another question for you to try.

Pause the video to complete the task, resume the video once you're finished.