Compute $\sigma$, given $\sigma^{11}$ in $S_{10}.$ 
Let $\sigma \in S_{10}$ with $\sigma^{11}=(3,1,4)(5,2,9,6,8,7,10)$. How to compute $\sigma$?

I tried:
$\sigma^{11}=(3,1,4)(5,2,9,6,8,7,10)=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 4 & 9 & 1 & 3 & 2 & 8 & 10& 7 & 6 & 5\end{pmatrix}$
Now I tried to find two permutations such that 
$\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\  &  &  &  &  &  & &  &  & \end{pmatrix}\cdot \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\  &  &  &  &  &  & &  &  & \end{pmatrix}=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 4 & 9 & 1 & 3 & 2 & 8 & 10& 7 & 6 & 5\end{pmatrix}$
But it's not working.
How to find $\sigma$ here?

Let $\tau= (314)(529687a)$ (with $a=10$) & so $\tau^{21}=e$ & $\sigma^{21}=e$. We have $ \tau= \sigma^{11}$. Squaring gives $\sigma=\tau^2$. Which is easy to compute.
