Question

Упростите выражения:

Answer 1

1) $\sqrt{\sqrt{28-16\sqrt3}}=\sqrt{\sqrt{12-2*2\sqrt3*4+16}}=\sqrt{|2\sqrt3-4|}=[2\sqrt3=\sqrt{12}<\sqrt{16}=4]=\sqrt{4-2\sqrt3}=\sqrt{3-2\sqrt3+1}=\sqrt3-1$

2) $\sqrt{2+\sqrt{2+\sqrt{2+\sqrt3}}}\sqrt{2-\sqrt{2+\sqrt{2+\sqrt3}}}=\sqrt{(2+\sqrt{2+\sqrt{2+\sqrt3}})(2-\sqrt{2+\sqrt{2+\sqrt3}})}=\sqrt{4-(2+\sqrt{2+\sqrt3})}=\sqrt{2-\sqrt{2+\sqrt3}}$$\sqrt{2+\sqrt3}\sqrt{2+\sqrt{2+\sqrt3}}\sqrt{2+\sqrt{2+\sqrt{2+\sqrt3}}}\sqrt{2-\sqrt{2+\sqrt{2+\sqrt3}}}=\\ \sqrt{2+\sqrt3}\sqrt{2+\sqrt{2+\sqrt3}}\sqrt{2-\sqrt{2+\sqrt3}}=\sqrt{2+\sqrt3}\sqrt{(2+\sqrt{2+\sqrt3})(2-\sqrt{2+\sqrt3})}=\sqrt{2+\sqrt3}\sqrt{(4-(2+\sqrt3)}=\sqrt{2+\sqrt3}\sqrt{2-\sqrt3}=\sqrt{4-3}=1$