Question

Find the value of $\sqrt{-\sqrt{3}+\sqrt{3+8\sqrt{7+4\sqrt{3}}}}$

Answer 1

given, $\sqrt{-\sqrt{3}+\sqrt{3+8\sqrt{7+4\sqrt{3}}}}$

$=\sqrt{-\sqrt{3}+\sqrt{3+8\sqrt{4+3+2\times 2\times \sqrt{3}}}}$

$=\sqrt{-\sqrt{3}+\sqrt{3+8\sqrt{(2{)}^2+(\sqrt{3}{)}^2+2\times 2\times \sqrt{3}}}}$

$=\sqrt{-\sqrt{3}+\sqrt{3+8\sqrt{(2+\sqrt{3}{)}^2}}}$

$=\sqrt{-\sqrt{3}+\sqrt{3+8(2+\sqrt{3})}}$

$=\sqrt{-\sqrt{3}+\sqrt{3+16+8\sqrt{3}}}$

$=\sqrt{-\sqrt{3}+\sqrt{(\sqrt{3}{)}^2+(4{)}^2+2\times 4\times \sqrt{3}}}$

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

$=\sqrt{-\sqrt{3}+4+\sqrt{3}}$

$=\sqrt{4}=2$