If $\beta = (172543)$, calculate $\beta^8$

Question

Please would someone check my answer?

My answer is $(124)(375).$ My method was to find $((\beta^2)^2)^2$.

Answer 1

You 're correct, but an easier way is to notice that $b$ is a 6-cycle, thus $b^6=1$ and thus $b^8=b^6b^2=b^2= (124)(375)$