cube of a number equals sixty-four times the same number

Paola, we write this as x^3=64x. Putting both on the same side, we get x^3-64x=0. Factoring, we get

x(x^2-64)=0

So, now we know that, because anything times 0 is 0, that x can equal 0, but we have to deal with the other part.

x^2-64=0

We have the difference of two squares. so (x-8)(x+8)=0

x-8=0, so x=8

x+8=0, so x=-8.

So, our final answer is that x=0, x=8 or x=-8.