Answer
$x=1$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given radical equation, $
\sqrt[3]{2x+5}=\sqrt[3]{6x+1}
,$ raise both sides of the equation to the third power. Then use properties of equality to isolate and solve the variable. Finally, do checking of the solution/s with the original equation.
$\bf{\text{Solution Details:}}$
Raising both sides of the equation to the third power results to
\begin{array}{l}\require{cancel}
\left(\sqrt[3]{2x+5}\right)^3=\left(\sqrt[3]{6x+1}\right)^3
\\\\
2x+5=6x+1
.\end{array}
Using the properties of equality to isolate the variable results to
\begin{array}{l}\require{cancel}
2x-6x=1-5
\\\\
-4x=-4
\\\\
x=\dfrac{-4}{-4}
\\\\
x=1
.\end{array}
Upon checking, $
x=1
$ satisfies the original equation.