Question

Let a,b,c and x elements in the group G. In each of the following solve for x in terms of a,b, and c.

solve simultaneously x^2 =a and x^5 = e

Answer 1

Every $g\in G$ has an inverse $g^{-1}\in G$ such that $gg^{-1}=g^{-1}g=e$. So

$x^5=x^2x^2x=e\implies a^2x=e\implies x=(a^{-1})^2$