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Simplify the nested radical.

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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,712
Simplify \(\displaystyle \sqrt{1+a^2+\sqrt{1+a^2+a^4}}\).
 

Albert

Well-known member
Jan 25, 2013
1,225
Simplify \(\displaystyle \sqrt{1+a^2+\sqrt{1+a^2+a^4}}\).
let :
\(\displaystyle \sqrt{1+a^2+\sqrt{1+a^2+a^4}}=\sqrt x +\sqrt y\).
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
$x=\dfrac {a^2-a+1}{2}$
$y=\dfrac {a^2+a+1}{2}$
$\therefore \,\, \sqrt{1+a^2+\sqrt{1+a^2+a^4}}=\sqrt{\dfrac{a^2+a+1}{2}}+\sqrt{\dfrac{a^2-a+1}{2}}$