Answer
$\{16\}$.
Work Step by Step
The given equation is
$\Rightarrow \sqrt {\sqrt x+\sqrt{x+9}}=3$
Square both sides.
$\Rightarrow \left (\sqrt {\sqrt x+\sqrt{x+9}}\right )^2=(3)^2$
Simplify.
$\Rightarrow \sqrt x+\sqrt{x+9}=9$
Subtract $\sqrt{x+9}$ from both sides.
$\Rightarrow \sqrt x+\sqrt{x+9}-\sqrt{x+9}=9-\sqrt{x+9}$
Simplify.
$\Rightarrow \sqrt x=9-\sqrt{x+9}$
Square both sides.
$\Rightarrow (\sqrt x)^2=(9-\sqrt{x+9})^2$
Use the special formula $(A+B)^2=A^2+2AB+B^2$
We have $A=9$ and $B=\sqrt{x+9}$
$\Rightarrow x=(9)^2-2(9)( \sqrt{x+9})+(\sqrt{x+9})^2$
Simplify.
$\Rightarrow x=81-18 \sqrt{x+9}+x+9$
Add $18 \sqrt{x+9} -x$ to both sides.
$\Rightarrow x+18 \sqrt{x+9} -x=81-18 \sqrt{x+9}+x+9+18 \sqrt{x+9} -x$
Simplify.
$\Rightarrow 18 \sqrt{x+9} =90$
Divide both sides by $18$.
$\Rightarrow \frac{18 \sqrt{x+9}}{18} =\frac{90}{18}$
Simplify.
$\Rightarrow \sqrt{x+9} =5$
Square both sides.
$\Rightarrow (\sqrt{x+9})^2 =(5)^2$
Simplify.
$\Rightarrow x+9 =25$
Subtract $9$ from both sides.
$\Rightarrow x+9-9 =25-9$
Simplify.
$\Rightarrow x =16$
The solution is $x=\{16\}$.
