#### Answer

$m=14$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given radical equation, $
\sqrt[3]{2m-1}=\sqrt[3]{m+13}
,$ 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]{2m-1} \right)^3=\left( \sqrt[3]{m+13} \right)^3
\\\\
2m-1=m+13
.\end{array}
Using the properties of equality to isolate the variable results to
\begin{array}{l}\require{cancel}
2m-m=13+1
\\\\
m=14
.\end{array}
Upon checking, $
m=14
$ satisfies the original equation.