Let G be a group and recall that the non-negative powers of an element x in G are defined as follows: Let G be a group and recall that the non-negative powers of an element x in G
are defined as follows: $x^{0} = e, x^{1} = x, $and $x^{n+1} = x^{n}x$ for n > 0. Now, let x and y be elements in G and, using induction, show that $ (x y x^{-1})
^{n} = x y^{n} x^{-1}$ holds for $ n > 0.$
can someone help me start this please

Hint: $$(xyx^{-1})^2=(xyx^{-1})(xyx^{-1})$$

Base case: when $n=1$, we have of course $$(xyx^{-1})=xyx^{-1},$$
which is true. For the inductive step, assume it's true for $n=k$, that is
$$(xyx^{-1})^k=xy^kx^{-1},$$ and let's see what happens for when $n=k+1$. We have $$(xyx^{-1})^{k+1}=xyx^{-1}(xyx^{-1})^k=xyx^{-1}xy^kx^{-1}=xyy^kx^{-1}=xy^{k+1}x^{-1},$$
which is precisely what we wanted to prove.