[tex]\large\underline{\sf{Solution-}}[/tex]
Given that,
[tex]\sf\sqrt{5+2\sqrt6}[/tex]
We can rewrite it as,
[tex]\sf\longmapsto\sqrt{5+2(\sqrt2)(\sqrt3)}[/tex]
[tex]\sf\longmapsto\sqrt{2+3+2(\sqrt2)(\sqrt3)}[/tex]
So,
[tex]\sf\longmapsto\sqrt{(\sqrt2)^2+(\sqrt3)^2+2(\sqrt2)(\sqrt3)}[/tex]
We know that,
[tex]\sf\red⇛ a^2+b^2+2ab=(a+b)^2[/tex]
Here, a = √2 and b=√3.
So,
[tex]\sf\longmapsto\sqrt{(\sqrt2)^2+(\sqrt3)^2+2(\sqrt2)(\sqrt3)}[/tex]
[tex]\sf\longmapsto\sqrt{(\sqrt2+\sqrt3)^2}[/tex]
So, the square root and the square get cancelled. So,
[tex]\sf\longmapsto\sqrt{(\sqrt2+\sqrt3)^2}[/tex]
[tex]\sf\longmapsto\pm\sqrt2+\sqrt3[/tex]
Therefore,
[tex]\bf\sqrt{5+2\sqrt6}=\pm\sqrt2+\sqrt3[/tex]
