Answer
$\left\{-1\right\}$
Work Step by Step
Expressing both sides of the given equation, $
5^{x+3}=\left(\dfrac{1}{25}\right)^{3x+2}
$, in the same base, then
\begin{align*}
5^{x+3}&=\left(\dfrac{1}{5^2}\right)^{3x+2}
\\\\
5^{x+3}&=\left(5^{-2}\right)^{3x+2}
&(\text{use }\dfrac{1}{a^m}=a^{-m})
\\\\
5^{x+3}&=5^{-2(3x+2)}
&(\text{use }\left(a^m\right)^n=a^{mn})
\\
5^{x+3}&=5^{-6x-4}
.\end{align*}
Since $x^a=x^b$ implies $a=b$, then the equation above implies
\begin{align*}
x+3&=-6x-4
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}\require{cancel}
x+6x&=-4-3
\\
7x&=-7
\\\\
\dfrac{\cancel7x}{\cancel7}&=-\dfrac{7}{7}
\\\\
x&=-1
.\end{align*}
Hence, the solution set of the equation $
5^{x+3}=\left(\dfrac{1}{25}\right)^{3x+2}
$ is $
\left\{-1\right\}
.$
