# Math Help - Solve the equation

1. ## Solve the equation

Solve the equaition ( $x\in\mathbb{R}$):
$\frac 1{\left(1+\sqrt{1+\sqrt{x}}\right)^4}+\frac 1{\left(1-\sqrt{1+\sqrt{x}}\right)^4}+\frac 2{\left(1+\sqrt{1+\sqrt{x}}\right)^3}+\frac 2{\left(1-\sqrt{1+\sqrt{x}}\right)^3}=0$

2. Hmm.. The solution is too long for me to write here, but let me express it. Let's call $\sqrt{1+\sqrt{x}} = a$

Then the equation is,
$\frac 1{\left(1+a\right)^4}+\frac 1{\left(1-a\right)^4}+\frac 2{\left(1+a\right)^3}+\frac 2{\left(1-a\right)^3}=0
$

What you must do here is to equalize the denominators by expanding all the fractions as they'll all have the denominator $\left(1+a\right)^4$
$\left(1-a\right)^4$

And then you can discard the common denominator, make some factorizations, simplify it as much as you can. Then this will give you a fourth degree equation which can be expressed as a quadratic equation. This will give you $a = \sqrt{1+\sqrt{\frac{8}{5}}}$ and you can get to $x = \frac{8}{5}$