power of five

Answer: $x=3$

Rearranging the equation$$\begin{align} 5\left(5^x\times5\right)^2&=5^9\\ 

\Rightarrow \left(5^x\times5\right)^2&=5^8\\

\Rightarrow 5^x\times5&=\sqrt{5^8}\\

&=5^4\\

\Rightarrow 5^x&=5^3\\

\Rightarrow x&=3\end{align}$$

Using index laws first

First, look at what is contained inside the brackets:$$\begin{split}5^x+5^x+5^x+5^x+5^x&=5\times5^x\\

&=5^1\times5^x\\

&=5^{1+x}\end{split}$$ So: $$\begin{split}5\left(5^x+5^x+5^x+5^x+5^x\right)^2&=5\left(5^{1+x}\right)^2\\

&=5\times5^{(1+x)\times2}\\

&=5^1\times5^{2+2x}\\

&=5^{1+2+2x}=5^{3+2x}\end{split}$$ So $5^{3+2x}=5^9$, which means $3+2x=9$, so $2x=6$, so $x=3$.