#### Answer

$x=25$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given equation, $
x^{3/2}=125
,$ raise both sides to the exponent equal to $
\dfrac{2}{3}
.$ Then use the laws of exponents and the definition of rational exponents to simplify the resulting equation. Finally, do checking if the solution satisfies the original equation.
$\bf{\text{Solution Details:}}$
Raising both sides to the exponent, $
\dfrac{2}{3}
,$ the given equation becomes
\begin{array}{l}\require{cancel}
\left(x^{\frac{3}{2}}\right)^{\frac{2}{3}}=(125)^{\frac{2}{3}}
.\end{array}
Using the Power Rule of the laws of exponents which is given by $\left( x^m \right)^p=x^{mp},$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
x^{\frac{3}{2}\cdot\frac{2}{3}}=125^{\frac{2}{3}}
\\\\
x=125^{\frac{2}{3}}
.\end{array}
Using the definition of rational exponents which is given by $a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
x=\left( \sqrt[3]{125} \right)^2
\\\\
x=\left( 5 \right)^2
\\\\
x=25
.\end{array}
Upon checking, $
x=25
$ satisfies the original equation.