13. Find the sum of all the real numbers $x$ that satisfy the equation
$$
\left(3^{x}-27\right)^{2}+\left(5^{x}-625\right)^{2}=\left(3^{x}+5^{x}-652\right)^{2} .
$$
13. Ans: 7
Let $a=3^{x}-27$ and $b=5^{x}-625$, so that the equation becomes
$$
a^{2}+b^{2}=(a+b)^{2} \Longrightarrow 2 a b=0 \Longrightarrow a=0 \text { or } b=0 \text {. }
$$

When $a=0$, we have $3^{x}-27=0 \Longrightarrow x=3$.
When $b=0$, we have $5^{x}-625=0 \Longrightarrow x=4$.
Hence, the sum is $3+4=7$.