Power of five

Answer: $x=3$

Rearranging the equation$$\begin{array}{rl}5{(5^x\times 5)}^2& =5^9\\ \Rightarrow {(5^x\times 5)}^2& =5^8\\ \Rightarrow 5^x\times 5& =\sqrt{5^8}\\ & =5^4\\ \Rightarrow 5^x& =5^3\\ \Rightarrow x& =3\end{array}$$

Using index laws first

First, look at what is contained inside the brackets:$$\begin{array}{rl}{5}^x+5^x+5^x+5^x+5^x& =5\times 5^x\\ & =5^1\times 5^x\\ & =5^{1+x}\end{array}$$ So: $$\begin{array}{rl}5{(5^x+5^x+5^x+5^x+5^x)}^2& =5{\left(5^{1+x}\right)}^2\\ & =5\times 5^{(1+x)\times 2}\\ & =5^1\times 5^{2+2x}\\ & =5^{1+2+2x}=5^{3+2x}\end{array}$$ So $5^{3+2x}=5^9$, which means $3+2x=9$, so $2x=6$, so $x=3$.