Answer
$6$.
Work Step by Step
The given expression is
$=(\sqrt{2+\sqrt3}+\sqrt{2-\sqrt3})^2$
Use the special formula $(A+B)^2=A^2+2AB+B^2$.
We have $A=(\sqrt{2+\sqrt3})$ and $B=(\sqrt{2+\sqrt3})$.
$=(\sqrt{2+\sqrt3})^2+2(\sqrt{2+\sqrt3})(\sqrt{2-\sqrt3})+(\sqrt{2-\sqrt3})^2$
Use product rule.
$=2+\sqrt3+2\sqrt{(2+\sqrt3)(2-\sqrt3)}+2-\sqrt3$
Use the special formula $(A+B)(A-B)=A^2-B^2$ and add like terms.
We have $A=2$ and $B=\sqrt 3$.
$=4+2\sqrt{(2)^2-(\sqrt3)^2}$
Use product rule.
$=4+2\sqrt{4-3}$
Simplify.
$=4+2$
$=6$.
