Hello Pharod
Welcome to Math Help Forum! Originally Posted by
Pharod The problem is the following:
$\displaystyle \sqrt{8+2\sqrt{10+2\sqrt{5}}}+\sqrt{8-2\sqrt{10+2\sqrt{5}}}$
I have to simplify it, but I have no idea how to start. Any help is appreciated. Thanks!
Pharod
Let $\displaystyle x = \sqrt{a+b}+\sqrt{a-b}$, where $\displaystyle a = 8$ and $\displaystyle b = 2\sqrt{10+2\sqrt5}$
Then $\displaystyle x^2 = a+b+2\sqrt{a^2-b^2}+a-b$ $\displaystyle =2a+2\sqrt{a^2-b^2}$
$\displaystyle =16+2\sqrt{24-8\sqrt5}$
$\displaystyle =4(4+\sqrt{6-2\sqrt5})$
Also $\displaystyle 6-2\sqrt5 = (\sqrt5 - 1)^2$
$\displaystyle \Rightarrow \sqrt{6-2\sqrt5} = \sqrt5 - 1$
$\displaystyle \Rightarrow x^2 = 4(4+\sqrt5-1)$
$\displaystyle \Rightarrow x = 2\sqrt{3+\sqrt5}$
I don't think it will simplify any more.
Grandad