First, the question doesn't even make sense if $\alpha$ is not an integer, because $W^\alpha$ is undefined as a real valued process when W goes negative. So, in that case, I'll assume that W is started from a positive value and stopped when it hits zero.
There's two methods: Use Ito's formula for $X=W^\alpha$,
$$
dX = \alpha W^{\alpha-1}\,dW + \frac12\alpha(\alpha-1) W^{\alpha-2}\,dt.
$$
The first term on the right is a local martingale. So the only way that X can be a martingale is for $\alpha(\alpha-1)\int W_t^{\alpha-2}\,dt$ to be a local martingale. It is a standard result that continuous finite variation processes cannot be a local martingale unless they are constant, giving $\alpha(\alpha-1)=0$.
Or, use Jensen's inequality. As $f(x)=x^\alpha$ is strictly convex for $\alpha\not\in[0,1]$ and strictly concave for $\alpha\in(0,1)$ (restricting to the positive reals), the function
$$t\mapsto\mathbb{E}[W_t^\alpha]$$
is strictly increasing for $\alpha\not\in[0,1]$ and strictly decreasing for $\alpha\in(0,1)$. So, $W^\alpha$ is not a martingale in these cases. In fact, it is a submartingale for $\alpha \not\in[0,1]$ and a supermartingale for $\alpha\in(0,1)$.
In this case, Jensen's inequality is the simplest method. However, Ito's formula applies much more generally, to any twice differentiable function of a semimartingale.