Answer
The solution is $y=2$.
Work Step by Step
The given equation is
$\Rightarrow \sqrt[3]{y+6}=\sqrt[3]{5y-2}$
Cube each side of the equation.
$\Rightarrow (\sqrt[3]{y+6})^3=(\sqrt[3]{5y-2})^3$
Simplify.
$\Rightarrow y+6=5y-2$
Add $2-y$ to each side.
$\Rightarrow y+6+2-y=5y-2+2-y$
Simplify.
$\Rightarrow 8=4y$
Divide each side by $4$.
$\Rightarrow 2=y$
Check $y=2$.
$\Rightarrow \sqrt[3]{y+6}=\sqrt[3]{5y-2}$
$\Rightarrow \sqrt[3]{2+6}=\sqrt[3]{5(2)-2}$
$\Rightarrow \sqrt[3]{8}=\sqrt[3]{10-2}$
$\Rightarrow \sqrt[3]{8}=\sqrt[3]{8}$
True.
Hence, the solution is $y=2$.
