Powers and Roots of Group Elements Let $G$ be a group, and $a$, $b \in G$ 


*

*$(bab^{-1})^{n} = ba^{n}b^{-1}$, for every positive integer $n$
\begin{align*}
\text{Let P(n) be the statement: } (bab^{-1})^{n} &= ba^{n}b^{-1} \newline
\text{Show the base case P(1) : } bab^{-1} &= bab^{-1} \newline
\text{Assume P(k) is true : } (bab^{-1})^{k} &= ba^{k}b^{-1} \newline
\text{Now I need to show P(k+1) is true by multiplying P(k) by } bab^{-1} \newline
(bab^{-1})^{k}(bab^{-1}) &= ba^{k}b^{-1}(bab^{-1}) \newline
&= ba^{k}(b^{-1}b)ab^{-1} \newline
&= ba^{k}ab^{-1} \newline
&= ba^{k+1}b^{-1} \newline
\text{Which is the statement P(k+1)}
\end{align*}

*If $a^{-1}$ has a cube root, so does $a$. 
If $a^{-1}$ does have a cube root, then there is an element $x$ in $G$ such that $a^{-1} = x^{3}$. 
\begin{align*}
a^{-1} &= x^{3} \newline
a^{-1}a &= x^{3}a \newline
e &= x^{3}a \newline
(x^{-1})^{3} &= (x^{-1})^{3}x^{3}a \newline
(x^{-1})^{3} &= a
\end{align*}
Comments on the correctness or ways to improve either proof would be appreciated :)

HINT $\ \ \ $ Both answers follow from $\rm\ f(x^n) = f(x)^n\ $ for an (anti-) multiplicative map $\rm\:f\:.$
Slightly simpler and more general is to note that the map $\rm\ f(x) = b\:x\:b^{-1}\ $ is multiplicative, i.e.  $\rm\:f(xy) = f(x)\:f(y)\:,\:$ so, by induction $\rm\ f(x^n) = f(x)^n\:.$
Similarly for the second problem $\rm\: f(x) = x^{-1}\: $ satisfies $\rm\ f(xy) = f(y)\:f(x)\:,\: $ therefore upon  applying $\rm\ f\ \:$ to $\rm\ a^{-1}\! =\: x^n\ $ we infer that $\rm\ a\: =\: f(x^n) = f(x)^n = (x^{-1})^n\ \ $ (for $\rm\:n=3\:$ in your case).
