Answer
$\color{blue}{\left\{-29, 35\right\}}$
Work Step by Step
The numerator of the rational exponent is even so raising both sides to $\frac{5}{2}$ will not work.
Raise both sides to the fifth power to obtain:
$((x-3)^{2/5})^5=4^5
\\(x-3)^2=1024$
Take the square root of both sides to obtain:
$\sqrt{x-3)^2}=\pm \sqrt{1024}
\\x-3 = \pm \sqrt{32^2}
\\x-3 = \pm 32$
Add $3$ to both sides of the equation to obtain:
$\\x=3\pm 32
\\x_1=3-32=-29
\\x_2=3+32=35$
Upon checking both proposed solutions satisfy the original equation.
Thus, the solutions set is $\color{blue}{\left\{-29, 35\right\}}$.
