proving no primitive roots exist modulo $2^n$ for n $\geq$ 3

Question

Ive been asked to prove by induction that no primitive roots exist modulo $2^n$ for n $\geq$ 3.

I have proven true for base case $n=3$, and assumed to be true for $n$. I'm now stuck at this point:

$${x^{2}}^{n-1} \equiv 1 \pmod{2^{n+1}}.$$

thank you :)