Question

Rút gọn biểu thức

\(\sqrt{2+\sqrt{3}}.\sqrt{2+\sqrt{2+\sqrt{3}}}.\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}.\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}\)

Answer 1

\(\sqrt{2+\sqrt{3}}.\sqrt{2+\sqrt{2+\sqrt{3}}}.\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}.\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}\)

\(=\sqrt{2+\sqrt{3}}.\sqrt{2+\sqrt{2+\sqrt{3}}}.\sqrt{4-\left(2+\sqrt{2+\sqrt{3}}\right)}\)

\(=\sqrt{2+\sqrt{3}}.\sqrt{2+\sqrt{2+\sqrt{3}}}.\sqrt{2-\sqrt{2+\sqrt{3}}}\)

\(=\sqrt{2+\sqrt{3}}.\sqrt{4-\left(2+\sqrt{3}\right)}\)

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

Answer 2

Rút gọn biểu thức

$\sqrt{2+\sqrt{3}}.\sqrt{2+\sqrt{2+\sqrt{3}}}.\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}.\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}$

= √$\sqrt{2+\sqrt{3}}.\sqrt{2+\sqrt{2+\sqrt{3}}}.\sqrt{4-\left(2+\sqrt{2+\sqrt{3}}\right)}$

$=\sqrt{2+\sqrt{3}}.\sqrt{2+\sqrt{2+\sqrt{3}}}.\sqrt{2-\sqrt{2+\sqrt{3}}}$

$=\sqrt{2+\sqrt{3}}.\sqrt{4-\left(2+\sqrt{3}\right)}$

$=\sqrt{2+\sqrt{3}}.\sqrt{2-\sqrt{3}}=4-3=1$