#### Answer

$z=5$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given radical equation, $
\sqrt[4]{z+11}=\sqrt[4]{2z+6}
,$ raise both sides of the equation to the exponent equal to the index of the radical. 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 fourth power results to
\begin{array}{l}\require{cancel}
\left( \sqrt[4]{z+11} \right)^4=\left( \sqrt[4]{2z+6} \right)^4
\\\\
z+11=2z+6
.\end{array}
Using the properties of equality to isolate the variable results to
\begin{array}{l}\require{cancel}
z-2z=6-11
\\\\
-z=-5
\\\\
z=\dfrac{-5}{-1}
\\\\
z=5
.\end{array}
Upon checking, $
z=5
$ satisfies the original equation.