\(f(x) := x^C\) is an odd function when \(C\) is odd, or an even function when \(C\) is even, and is monotonically increasing in \(x \geq 0\) in both cases, so the ordering of \(f(A)\) and \(f(B)\) is equal to that of \(A\) and \(B\) if \(C\) is odd, or that of \(|A|\) and \(|B|\) if \(C\) is even.