Video: The Relation between Square Roots and Cube Roots

Determine the square of the number whose cube is −512.

02:39

Video Transcript

Determine the square of the number whose cube is negative 512.

Before we try answering this question, let’s quickly remind ourselves of what a square number is and what a cube number is. Starting with a cube number, we could write this as 𝑥 to the third power or 𝑥 cubed. And it is equivalent to 𝑥 times 𝑥 times 𝑥. So, for example, if we had two cubed, this is the same as two times two times two. Evaluating this, we would work out two times two, which is four, and then multiply it by the final two. Since four times two is eight, then two cubed is equal to eight. For a square number such as 𝑥 squared, this is equivalent to 𝑥 times 𝑥. For example, four squared can be written as four times four, which is 16.

In our question, we have a cube number that is a negative number. Let’s see how we might get a negative value for a cube member. If we imagine we have negative two cubed, this would be equivalent to negative two times negative two times negative two. Multiplying the first two negative twos would give us a positive four. And then multiplying by the final negative two would give us negative eight.

So if we have a value for a cube number that is a negative value, this means that the number that we cube or the cube root must also be negative. So let’s try some negative numbers cubed and see if we can find negative 512. If we had negative one cubed, that’s the same as negative one times negative one times negative one, which is negative one. Negative two cubed will give us negative eight. Negative three cubed will give us negative 27.

We can work our ways with the next few negative values. And we notice that negative eight cubed will give us negative 512. So we’re part away through answering the question. We found the number whose cube is negative 512. But now we need to find the square of this number. So we take our negative eight and we square it, giving us negative eight times negative eight, which is 64. Noticing here that our negative times a negative will give us our positive value, 64, and our final answer for the square of the number whose cube is negative 512.