Answer
$x=16$
Work Step by Step
Raising both sides to the second power of the given equation, $
\sqrt[]{x+9}=2+\sqrt[]{x-7}
,$ then the solution/s is/are
\begin{array}{l}\require{cancel}
x+9=(2+\sqrt[]{x-7})^2
\\\\
x+9=(2)^2+2(2)(\sqrt[]{x-7})+(\sqrt[]{x-7})^2
\\\\
x+9=4+4\sqrt[]{x-7}+x-7
\\\\
(x-x)+(9-4+7)=4\sqrt[]{x-7}
\\\\
12=4\sqrt[]{x-7}
\\\\
\dfrac{12}{4}=\dfrac{4\sqrt[]{x-7}}{4}
\\\\
3=\sqrt[]{x-7}
.\end{array}
Squaring both sides again, then,
\begin{array}{l}\require{cancel}
9=x-7
\\\\
9+7=x
\\\\
16=x
\\\\
x=16
.\end{array}
Upon checking, $
x=16
$ satisfies the original equation.