proving something implies another by contradiction

I have a question on a problem i am working on. I know this problem is
easier working with contrapositive and I can prove it that way but I
rather work with contradiction because its more challenging.

I am asked to
Prove that if x^3 is irrational, x is irrational.

my first problem is representing this as a contradiction statement. I
am not sure if it can be. Usually when we do contradiction proof is
usually one statement ie. prove this is irrational, no implications
involved.

My representation of this statement as a contradiction is to suppose
that x^3 is a rational and and that x should still remain irrational

going from there I say if x^3 is rational then it can be represented
by two whole integer a/b such that both are not equal to 0 and have
their gcd=1
 x^3 = a/b
x=(a/b)^3
x=(a/b)
since x= can be represented a/b is it a contradiction to our statement
that x is irrational
QED

Is this correct? Or did i represent my contradiction statement
incorrectly?
Thanks for clarification.