I second e.mccormick's inquiry. I know computer science does proofs, but the best section would be math! They have more people who are good at this sort of thing. This problem is mathematical in nature, anyway. Also, when people want to go through old math questions, they'll miss this if it's in computer science.
Your proof is great! It's perfect for an intro to proof, basic assignment, and two changes will make it better.
Firstly, show all the steps (basic steps are important in basic proofs) and make sure that you say that you are using the axiom and "definition of powers" or something like that.
Secondly, get from \((-x)^3\) all the way to \(-(x^3)\). It isn't a problem here, which I could prove. Generally, just using things equal to each other, it's not important. But it's a good habit to show that you start with \((-x)^3\) and, using just that and axioms, get to \(-(x^3)\).
You'll need to write out sentences, but here are my steps. Note that I took your given "-x = (-1)*x" and changed it to "-a = (-1)*a". I did this because we're not talking about the same variable, so we shouldn't use the same name. The \(a\) is arbitrary in \(\sf R\). The \(a\) doesn't need to be \(x\). I used \(a=x^3\) at one point.
Given (supposed to be true)
\((-x)^3\)
\(\downarrow\)
Definition of powers
\(=(-x)(-x)(-x)\)
\(\downarrow\)
-a = (-1)*a (Given) where \(a=x\)
\(=(-1)(x)(-1)(x)(-1)(x)\)
\(\downarrow\)
(No need to say anything)
\(=(-1)(x)(x)(x)\)
\(\downarrow\)
Definition of powers
\(=(-1)(x^3)\)
\(\downarrow\)
-a = (-1)*a (Given) where \(a=x^3\)
\(=-(x^3)\)
\(\downarrow\)
Transitive property of equivalence relations (you might not be required to write this out, it depends on your teacher.
\((-x)^3=-(x^3)\)\(\quad\Large\checkmark\)
I'm pretty certain about everything I said, but it might benefit you to ask this is in the math section for verification!