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

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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.