Video Transcript
Complete the following: The cube root of eight is equal to the sixth root of blank.
Let’s consider this problem by working out what the value on the left-hand side would be, the cube root of eight. Let’s say that we define the cube root of eight as some value 𝑎. And so if the cube root of eight is equal to 𝑎, then 𝑎 cubed is equal to eight. And so what is the value of 𝑎 such that 𝑎 times 𝑎 times 𝑎 gives us eight? Well, it must be two, since two times two is four and multiplied by another two would give us eight. Therefore, we know that two cubed is equal to eight. And so the cube root of eight must be two.
Once we’ve worked out this value of two for the left-hand side, we need to be careful. It can be a common mistake to think that this must be the missing answer. However, since we know that the left-hand side is equal to two, then in order to make this equation correct, everything on the right-hand side must also equal two. So now we have that the sixth root of something is equal to two. So let’s define this some value as 𝑏.
So in order to find this value of 𝑏, we can rearrange the equation. And we need to perform the inverse operation to finding the sixth root. So we would need to take the sixth power of two. Two to the sixth power is equivalent to six twos written and multiplied together. There are a number of different ways in which we can actually work this out. We could simply begin with two times two is four and then continue to multiply the result by two. So two times two is four, four times two is eight, eight times two is 16, 16 times two is 32, and 32 times two is 64.
Equivalently, we might also notice that two times two times two is eight. And we have two lots of those multiplied together. Eight times eight is 64. Either way that we’ve worked out that, the value of 𝑏, which we said is this unknown value within this sixth root, is 64. Therefore, we can give the answer as 64, since the cube root of eight is equal to the sixth root of 64.
