Answer
$\left\{0\right\}$
Work Step by Step
Expressing both sides of the given equation, $
5^{x+2}=25^{2x+1}
$, in the same base results to
\begin{align*}\require{cancel}
5^{x+2}&=\left(5^2\right)^{2x+1}
\\
5^{x+2}&=5^{4x+2}
.\end{align*}
Since $x^m=x^n$ implies $m=n$, the equation above implies
\begin{align*}\require{cancel}
x+2&=4x+2
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}\require{cancel}
2-2&=4x-x
\\
0&=3x
\\\\
\dfrac{0}{3}&=\dfrac{\cancel3x}{\cancel3}
\\\\
0&=x
.\end{align*}
Hence, the solution set of the equation $
5^{x+2}=25^{2x+1}
$ is $
\left\{0\right\}
$.
