Answer
$x=\dfrac{3}{2}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
(2x+5)^{1/3}-(6x-1)^{1/3}=0
,$ transpose first the second term to the right side and then raise both sides to the third power. Then use the properties of equality to combine like terms and to solve for the value of the variable. Finally, do checking if the solution satisfies the original equation.
$\bf{\text{Solution Details:}}$
Transposing the second term, the given equation becomes
\begin{array}{l}\require{cancel}
(2x+5)^{1/3}=(6x-1)^{1/3}
.\end{array}
Raising both sides to the third power, the equation above is equivalent to
\begin{array}{l}\require{cancel}
2x+5=6x-1
.\end{array}
Using the properties of equality, the solution to the equation above is
\begin{array}{l}\require{cancel}
2x-6x=-1-5
\\\\
-4x=-6
\\\\
x=\dfrac{-6}{-4}
\\\\
x=\dfrac{\cancel{-2}(3)}{\cancel{-2}(2)}
\\\\
x=\dfrac{3}{2}
.\end{array}
Upon checking, $
x=\dfrac{3}{2}
$ satisfies the original equation.
