$\alpha$ root of $x^{p^{d}} - x$ then it is a root of $x^{p^{ds}} - x$

Question

I want to prove that if $\alpha$ is a root of $x^{p^{d}} - x$ and $n=d\cdot s$, then $\alpha$ is also a root of $x^{p^{n}} - x$.
So far I know that $x^{p^{d}-1} =1$ and I've tried this: $$x^{p^{n} -1} = x^{p^{ds}-1}$$ I have also tried: $$x^{p^{n}} = x^{p^{ds}} = (x^{p^{d}})^s=\alpha^s$$ but I can't conclude that $\alpha^s=\alpha$. Can someone help me?