# Video: Pack 4 • Paper 3 • Question 7

Pack 4 • Paper 3 • Question 7

03:37

### Video Transcript

Part one) Find the value of the fourth root of 0.81 multiplied by 10 to the power of 14. Part two) Find the value of six to the power of two-thirds. Write your answer correct to two decimal places.

So the first thing we’re gonna to do for part one is actually rewrite what we’ve been given. So the way we’ve actually rewritten it is 81 over 100. And that’s because 0.81 is the same as eighty-one hundredths. So that will be 81 over 100 multiplied by 10 to the power of 14 and then this is all in brackets and it’s actually to the power of a quarter. And the reason it’s to the power of a quarter is as we’ve shown in the index law on the right-hand side, this is one of the things that we know that if we have 𝑥 to the power of one over 𝑎, then we’ll actually get the root to that same value of 𝑎.

Okay, so let’s now move on to the next stage. Well, what we can now say is this is gonna be equal to 81 multiplied by 10 to the power of 12 and this is all to the power of a quarter. And the reason is 81 multiplied by 10 to the power of 12 is- so looking back at the line before, we can see that we were dividing by 100 and then multiplying by 10 to the power of 14. Well, what we can actually do is divide 10 to the power of 14 by 100, which gives us 10 to the power of 12.

Okay, so as I said, we’ve got 81 multiplied by 10 to the power of 12 all to the power of a quarter. So now, what I’ve done is actually rewritten it again with three to the power of four multiplied by 10 to the power of 12 and this is all to the power a quarter. And that is cause three to the power of four is 81.

So now, what I’m gonna do is actually simplify further. And I can do that using another index law. And this index law tells me that 𝑥 to the power of 𝑎 then to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 multiplied by 𝑏. So now, what I’ve done is actually written one more line out that I wouldn’t usually write out, but just to show you what’s happening. So we’ve got three to the power of and then four multiplied by a quarter multiplied by 10 to the power of 12 multiplied by a quarter which therefore gives us an answer of three multiplied by 10 to the power of three. And that’s because four multiplied by a quarter is just one and 12 multiplied by a quarter is three.

So therefore, we can say that the value of the fourth root of 0.81 multiplied by 10 to the power of 14 is gonna be equal 3000. And that’s because three multiplied by 10 to the power of three is 3000. Okay, great, we’ve completed part one. Let’s move on to part two.

Well, in part two, to actually find a value of six to the power of two over three, there’s a couple of ways you can do this. If you’ve got a scientific calculator, you can actually just put this into your calculator and see what the result is. And when you do that, you’ll get 3.301 et cetera. And then, if you’d look back at the question, it tells you to write your answer correct to two decimal places. So therefore, we go after the second decimal place, which is a zero. And you will look at the number after it, which is a one. And because this one is actually less than five, the zero stays the same. So therefore, we can say that six to the power of two-thirds is equal to 3.30 to two decimal places.

And what mostly we’re gonna do now is show you kinda where it came from this answer. So for instance, if you didn’t have scientific calculator, how you might calculate it? Well, if you think about six to the power of two-thirds, this is equal to the cube root of six squared. And that’s actually partially using the index law that we looked at first.

So if we’re saying that it’s the cube root of six squared, then this straightforward would say that it’s the cube root of 36, which would then give us the same answer as we got using the first method. So we can say that the value of six to the power of two-thirds is 3.30 correct to two decimal places.