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Hi, my name is Mr. Chan.

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

Let's begin with some fraction multiplications that we should already know how to do.

So we're going to start off with one third multiplied by two, that would equal two thirds, one fifth, multiplied by four, that would equal four over five, one half, multiply by three, that equals three over two.

And finally one quarter multiply by seven, that would equal, seven over four.

So we can apply this to almost simplifying indices.

So here we've got a to the power of one third inside the bracket, and it's all squared.

So when we deal with indices inside the bracket, we would multiply those two index numbers together.

So the answer would be a to the power of one third multiply by two, which we already have done at the top there.

So a to the power of two thirds.

Similarly, b to the power of one fifth inside the bracket all to the power of four, we would multiply those two index numbers together.

So b to the power of one fifth multiplied by four would equal, b to the power of four fifths.

Similarly with C to the power of a half inside the bracket cubed, that would equal c to the power of three of two, and d to the power of one quarter inside the bracket to the power of seven, that would equal d to the power of seven over four.

So we can apply what I've just discussed in the previous examples into answering something like this.

We've got to calculate sixty four to the power of two thirds.

So sixty four to the power of two thirds, I can think of as, sixty four to the power of one third inside a bracket squared, because remember, we would multiply those two index numbers together, one third multiplied by two is the same as two thirds.

Now, what we've done with fractional indices previously, the power of one third means that we've just got a cube root.

So sixty four to the power of one third equals the cube root of sixty four.

Now that's still inside the bracket.

So if we figure out what the cube root of sixty four is, that would be four.

We're left with four squared, which we can finally give an answer four squared equals sixteen.

Let's try another one.

If we were to calculate one hundred to the power of three over two, we can think of that as, one hundred to the power of a half, to the power of three now the hundred to the power of half is inside the bracket.

That's the same as one hundred to the power of a half times three, which is one hundred to the power three over two, so, a number to the power of a half we know just the square root.

So that would be the square root of a hundred inside the bracket cubed.

The square root of one hundred is ten.

That leaves us with ten cubed, and ten cubed equals one thousand.

Here's a question you can try.

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

Hear the answers.

So when we have fractional indices like this, so for example, let's look at twenty five to the power of three over two, try and think of that fraction in the index number as a fraction multiplied by an integer.

So three over two is a fraction I can think of as half multiplied by three, that indicates that it would be twenty five to the power of a half in a bracket to the power of three outside the bracket.

So twenty five to the power of a half is the square root twenty five to the power of three, the square root of twenty five is five, so five to the power three would be one hundred and twenty five.

Here's another question you can try.

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

Here are the answers.

Let's review and look over part a, sixty four to the power of two thirds.

Now sixty four to the power of two thirds, I can think of as sixty four to the power of one third inside the bracket.

And outside of the bracket that would be squared, because multiplying those two index numbers together would be one third multiplied by two, which is the same as two thirds.

So the power of one third indicates a cube root.

So inside the bracket I would have the cube root of sixty four still all squared.

So the cube root of sixty four would be four.

That leaves us four square to answer four squared equals sixteen.

Here's some more questions you can try.

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

Here are the answers.

There are a couple of questions here with negative index values.

So if you aren't sure about those, it might be worth having a look at my lesson on negative indices.

Other than that, I hope you got all of these correct.

That's all for this lesson.

Thanks for watching.