jmc

It may be easier to work with this if we leave both sides in somewhat factored form: $$\begin{array}{rl}(5x{)}^4& =(10x{)}^3\\ \Rightarrow 5^4{x}^4& =10^3{x}^3\\ \Rightarrow 5^4{x}^4& =5^3{2}^3{x}^3\end{array}$$ Since $x$ is non-zero, we can cancel the common factor of $x^3$: $$\Rightarrow 5^{4}x=5^3{2}^3$$ Now, cancel $5^3$: $$\begin{array}{rl}5x& =8\\ \Rightarrow x& =\boxed{\frac{8}{5}}\end{array}$$