How do I prove that if x is odd then x^16 is odd? I know how to show $x^2$ is odd but unsure about x^16

Also assuming an odd number = 2k+1
Just apply your proof four times.
For example $x^2$ is odd gives $(x^2)^2=x^4$ is odd... and so on and so forth
If $x=2y+1$ then
$$
x^n=\sum_{i=0}^n \binom{n}{i}2^iy^i=2\left(\sum_{i=1}^n \binom{n}{i}2^{i-1}y^i\right)+1,
$$
which is odd.