Cardinality of $X^n$

Question

I asked the following question before: Is this proof by induction that $|\Bbb Q^n|=\aleph_0$ correct?

I want to know if the proccess I did that can be generalized to the case $|X|=\kappa; \kappa$ being any infinite cardinal.

This would be as follows:

Suppose you know $|X|=\kappa, |X \times X|=\kappa\cdot\kappa=\kappa$

Then proceed by induction, suppose $|X^n|=\kappa$, then $|X^{n+1}|=|X^n\times X|=\kappa\cdot\kappa=\kappa$.

Is this correct?

Answer 1

If $|X \times X|=|X|$, then $|X^{n}|=|X|$ by induction (your argument), so yes.