Answer
See below.
Work Step by Step
Proofs using mathematical induction consists of two steps:
1) The base case: here we prove that the statement holds for the first natural number.
2) The inductive step: assume the statement is true for some arbitrary natural number $n$ larger than the first natural number, then we prove that then the statement also holds for $n + 1$.
Hence here:
1) For $n=1$: if $x\gt1$, then $x^1=x\gt1$.
2) Assume for $n=k: x^k\gt1$. Then for $n=k+1$:
$x^{k+1}=x^k\cdot x$ and $x^k\gt1$ and $x\gt1$, thus their product will also be greater than $1$
Thus we proved what we wanted to.