Question

Prove for all x in R [(-x)^3 = -(x^3)]
(Hint you may use the fact that -x = (-1)*x, but other wise stick to axioms)
I wrote something along the lines of .
(-x)^3 can be written as (-1)(x)(x)(x) and -(x)^3 can be written as (-1)((x)(x)(x))
and these are both equivalent
But it doesn't feel like I'm proving anything.

Answer 1

Could start by rewriting (-x)^3 as (-x)(-x)(-x),
then show that this equals -x(x^2)
x^2 is always positive, and x(x^2) = x^3, so (-x)^3 = -(x^3).

Probably need a little more in there to make a formal rigorous proof out of it, but it's a possible starting point.