Complete the following: The fifth root of 243 is equal to the cube root of what.
The first thing to realize here is that we have a problem involving 𝑛th roots. On the left-hand side, we have the fifth root of 243. Notice that this five is written smaller within the root sign. It’s different to five multiplied by the square root of 243. This is not what we have here. We have a fifth root. So let’s approach this problem by seeing if we can work out what the left-hand side would be equal to.
In order to find the fifth root of 243, we could say that there’s some value — let’s call it 𝑥 — and when we take that to the fifth power, that would give us 243. Let’s consider what value 𝑥 could be. We know that it can’t be one since one to the power five would give us one. And 𝑥 couldn’t be two since any exponent of two will give us an even value. So let’s see if 𝑥 could be three. Does three to the power of five give us 243?
Well, three times three would give us nine. And then when we multiply that by the third three, that would give us 27. Then, multiplied by the fourth three, that would give us 81. And finally, 81 multiplied by the final three would indeed give us 243.
Now, we can say that since three to the power of five gives us 243, then the fifth root of 243 is three. That means that we’ve simplified the left-hand side of this equation. So let’s take a look at the right-hand side.
The blank that we’re missing is some value that when we take the cube root of it, it gives us three. In order to solve this equation, we’d need to perform the inverse operation. In this case, the inverse operation to taking the cube root is cubing or finding the third power. Three to the power of three is three times three times three, which gives us 27. Therefore, the missing value must be 27.
In order to check that this answer of 27 would be correct, we can remember that the fifth root of 243 is three. So that’s the left-hand side. And on the right-hand side, the cube root of 27 would also give us three. Since the two sides of the equation are equal, then 27 must be the correct answer.