How to simplify this expression?$$\sqrt{\smash[b]{18+\sqrt{260}}}-\sqrt{\smash[b]{12+\sqrt{140}}}-\sqrt{\smash[b]{20-2\sqrt{91}}},$$ which equals $0$. But how do I prove it?
My attempt
\begin{align}\small\sqrt{18+2\sqrt{65}} - \sqrt{12+2\sqrt{35}} - \sqrt{20-2\sqrt{91}}&\small= \sqrt{\left(\sqrt5 + \sqrt{13}\right)^2} - \sqrt{\left(\sqrt5+\sqrt7\right)^2}-\sqrt{\left(\sqrt{13}-\sqrt7\right)^2}\\&=\sqrt5+\sqrt{13}-\sqrt5-\sqrt7-\sqrt{13}+\sqrt7=0\end{align}
Pelle
] (18+2(65)^0.5)^0.5 - (12+2(35)^0.5)^0.5 - (20-2(91)^0.5)^0.5 [
] (5^0.5 + 13^0.5)^0.5 = 18 + 265^0.5 [
] (5^0.5 + 7^0.5)^0.5 = 12 + 235^0.5 [
] (7^0.5 - 13^0.5)^0.5 = 20 - 2*91^0.5 [
] 5^0.5 + 13^0.5 - 5^0.5 - 7^0.5 - 7^0.5 + 13^0.5 [
] 0 – Pelle Oct 21 '18 at 09:34