#### Answer

$x=35$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given equation, $
(x-3)^{2/5}=4
,$ raise both sides to the exponent equal to $
\dfrac{5}{2}
.$ Then use the laws of exponents and the concepts of rational exponents to solve the resulting equation. Finally, do checking if the solution satisfies the original equation.
$\bf{\text{Solution Details:}}$
Raising both sides to the exponent equal to $
\dfrac{5}{2}
,$ the given equation becomes
\begin{array}{l}\require{cancel}
\left( (x-3)^{2/5} \right)^{5/2}=4^{5/2}
.\end{array}
Using the Power Rule of the laws of exponents which is given by $\left( x^m \right)^p=x^{mp},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
(x-3)^{\frac{2}{5}\cdot\frac{5}{2} }=4^{\frac{5}{2}}
\\\\
x-3=4^{\frac{5}{2}}
.\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-3=\left(\sqrt{4}\right)^{5}
\\\\
x-3=\left(2\right)^{5}
\\\\
x-3=32
\\\\
x=32+3
\\\\
x=35
.\end{array}
Upon checking, $
x=35
$ satisfies the original equation.