MHB How to Simplify Nested Radicals?

[anemone]

Simplify $$\sqrt{1+a^2+\sqrt{1+a^2+a^4}}$$.

 

[Albert1]

Simplify $$\sqrt{1+a^2+\sqrt{1+a^2+a^4}}$$.

Spoiler

let :
$$\sqrt{1+a^2+\sqrt{1+a^2+a^4}}=\sqrt x +\sqrt y$$.
square both side we obtain :
$x+y=1+a^2----(1)$
$xy=\dfrac {a^4+a^2+1}{4}----(2)$
solving for (1)(2) we get :
$x=\dfrac {a^2+a+1}{2}$
$y=\dfrac {a^2-a+1}{2}$
or
$x=\dfrac {a^2-a+1}{2}$
$y=\dfrac {a^2+a+1}{2}$
$\therefore \,\, \sqrt{1+a^2+\sqrt{1+a^2+a^4}}=\sqrt{\dfrac{a^2+a+1}{2}}+\sqrt{\dfrac{a^2-a+1}{2}}$