#### Answer

{$57$}

#### Work Step by Step

Step 1: $\sqrt[3] {x+7}=4$
Step 2: Raising each side of the equation to the power of $3$ and simplifying, we find:
$(\sqrt[3] {x+7})^{3}=(4)^{3}$
$x+7=64$
Step 3: Solving the equation, we obtain:
$x+7=64$
$x=64-7$
$x=57$
Therefore, the solution set of the equation is {$57$}.