Alg2.1.29

Calculate $\sqrt{\sqrt{137-36\sqrt{14}}}\,$

Let

$137-36\sqrt{14}=137-2\sqrt{324\cdot14}\,$

$137-2\sqrt{4536}\,$

$81+56-2\sqrt{81\cdot56}\,$

$(9-\sqrt{56})^{2}\,$

Therefore

$\sqrt{137-36\sqrt{56}}=9-\sqrt{56}\,$

Now $\sqrt{9-\sqrt{56}}=\sqrt{7+2-2\sqrt{7\cdot2}}\,$

$\sqrt{7}-\sqrt{2}\,$

Hence

$\sqrt[4]{137-36\sqrt{14}}=\sqrt{7}-\sqrt{2}\,$