Exhibit a cycle with a power having a given cycle shape Problem: If $$τ=(1~ 2)(3 ~4)(5~ 6)(7~ 8)(9~ 10)$$ determine whether there is an $n$-cycle $σ,~~n≥10$ such that $τ=σ^k$ for some integer $k$.
I found let $σ=(1~ 3~ 5~ 7 ~9 ~2~ 4 ~6 ~8~ 10)$, then $$σ^5=(1~ 2)(3~ 4)(5 ~6)(7~ 8)(9~ 10)$$ But what is the process this is deduced? How this is derived?

A $10$-cycle has order $10$. $\tau$ has order $2$. So, if $\tau$ is a power of a $10$-cycle, then it must be the 5th power of that $10$-cycle. 
Now write down a general $10$-cycle, $(a\ b\ c\ d\ e\ f\ g\ h\ i\ j)$, and compute its 5th power, and compare with $\tau$ to see the possibilities for the $10$-cycle. 
