Alg2.1.16

Show that $\sqrt{2+\sqrt{5}-\sqrt{6-3\sqrt{5}+\sqrt{14-6\sqrt{5}}}}=2\,$

Simplifying the expression on the left side

$\sqrt{2+\sqrt{5}-\sqrt{6-3\sqrt{5}+\sqrt{9+5-2\sqrt{5\cdot9}}}}\,$

$\sqrt{2+\sqrt{5}-\sqrt{6-3\sqrt{5}+\sqrt{(\sqrt{9})^{2}+(\sqrt{5})^{2}-2\sqrt{5\cdot9}}}}\,$

$\sqrt{2+\sqrt{5}-\sqrt{6-3\sqrt{5}+(\sqrt{\sqrt{9}-\sqrt{5}})^{2}}}\,$

$\sqrt{2+\sqrt{5}-\sqrt{6-3\sqrt{5}+3-\sqrt{5}}}\,$

$\sqrt{2+\sqrt{5}-\sqrt{6-4\sqrt{5}+3}}\,$

$\sqrt{2+\sqrt{5}-\sqrt{6-2\sqrt{5\cdot4}+3}}\,$

$\sqrt{2+\sqrt{5}-\sqrt{5+4-2\sqrt{5\cdot4}}}\,$

$\sqrt{2+\sqrt{5}-\sqrt{(\sqrt{5})^{2}+(\sqrt{4})^{2}}}\,$

$\sqrt{2+\sqrt{5}-\sqrt{(\sqrt{5}-\sqrt{4})^{2}}}\,$

$\sqrt{2+\sqrt{5}-(\sqrt{5}-\sqrt{4}}\,$

$\sqrt{2+\sqrt{5}-\sqrt{5}+2}\,$

After simplifying we get

$\sqrt{4}\,$

Hence

$2\,$

Which is equal to right hand side