Evaluate the cube root of eight.
In this question, we need to be careful. This is the cube root of eight. It is not the same as three and then a square root of eight. This would be equivalent to three times the square root of eight. But this is not what we have in this question.
What we have here is in fact an 𝑛th root. And we can recall that the 𝑛th root 𝑟 of a quantity 𝑧 is a value such that 𝑧 is equal to 𝑟 to the power of 𝑛. The 𝑛th root can also be written as 𝑟 equals the 𝑛th root of 𝑧. And it’s this second statement which is more similar to what we have in the question.
So let’s say that we have some root 𝑟 which is equal to the cube root of eight. This would mean that eight is equal to 𝑟 cubed. And so now we must consider what value cubed does give us eight. Could it be one cubed? That would be one times one times one. Well, no, because one cubed would give us one, and not value of eight that we were looking for.
Next, we can ask, could it be two cubed? Well, that would be two times two times two. And two cubed does give us eight. We could say that eight is equal to two cubed. And that means that the cube root of eight is two. And so we’ve found the 𝑛th root, in this case the cube root, that we were looking for. It’s two.