a)\(\sqrt{5-2\sqrt{6}}\)
\(=\sqrt{3-2\sqrt{6}+2}\)
\(=\sqrt{3-2\sqrt{2}\sqrt{3}+2}\)
\(=\sqrt{\left(\sqrt{3}-\sqrt{2}\right)^2}\)
\(\left|\sqrt{3}-\sqrt{2}\right|\)
\(a,\sqrt{5-2\sqrt{6}}=\left(\sqrt{2}-\sqrt{3}\right)^2=|\sqrt{2}-\sqrt{3}|=\sqrt{3}-\sqrt{2}\)
\(b,\sqrt{5\sqrt{3}+5\sqrt{48-10\sqrt{7+4\sqrt{3}}}}\)
\(=\sqrt{5\sqrt{3}+5\sqrt{48-10\sqrt{\left(2+\sqrt{3}\right)^2}}}\)
\(=\sqrt{5\sqrt{3}+5\sqrt{48-10\left(2+\sqrt{3}\right)}}\)
\(=\sqrt{5\sqrt{3}+5\sqrt{48-\left(20-10\sqrt{3}\right)}}\)
\(=\sqrt{5\sqrt{3}+5\sqrt{28-10\sqrt{3}}}\)
\(=\sqrt{5\sqrt{3}+5\sqrt{\left(5-\sqrt{3}\right)^2}}\)
\(=\sqrt{5\sqrt{3}+5\left(5-\sqrt{3}\right)}\)
\(=\sqrt{5\sqrt{3}+25-5\sqrt{3}}\)
\(=\sqrt{25}=5\)
\(c,\sqrt{94-42\sqrt{5}}-\sqrt{94+42\sqrt{5}}\)
\(=\sqrt{\left(3\sqrt{5}-7\right)^2}-\sqrt{\left(3\sqrt{5}+7\right)^2}\)
\(=|3\sqrt{5}-7|-|3\sqrt{5}+7|\)
\(=7-3\sqrt{5}-3\sqrt{5}-7\)
\(=-6\sqrt{5}\)
\(\sqrt{5\sqrt{3}+5\sqrt{48-10\sqrt{7+4\sqrt{3}}}}\)
\(=\sqrt{5\sqrt{3}+5\sqrt{48-10\sqrt{2^2+4\sqrt{3}+3}}}\)
\(=\sqrt{5\sqrt{3}+5\sqrt{48-10\sqrt{\left(2+\sqrt{3}\right)^2}}}\)
\(=\sqrt{5\sqrt{3}+5\sqrt{48-10\left(2+\sqrt{3}\right)}}\)
\(=\sqrt{5\sqrt{3}+5\sqrt{48-20-10\sqrt{3}}}\)
\(=\sqrt{5\sqrt{3}+5\sqrt{28-10\sqrt{3}}}\)
\(=\sqrt{5\sqrt{3}+5\sqrt{5^2-2.5\sqrt{3}+3}}\)
\(=\sqrt{5\sqrt{3}+5\sqrt{\left(5-\sqrt{3}\right)^2}}\)
\(=\sqrt{5\sqrt{3}+5\left(5-\sqrt{3}\right)}\)
\(=\sqrt{5\sqrt{3}+25-5\sqrt{3}}\)
\(=\sqrt{25}\)
\(=5\)
\(A=\sqrt{94-42\sqrt{5}}-\sqrt{94+42\sqrt{5}}\)
\(A^2=\left(\sqrt{94-2.21\sqrt{5}}-\sqrt{94+42\sqrt{5}}\right)^2\)
\(A^2=94+42\sqrt{5}+94-42\sqrt{5}-2\sqrt{\left(\sqrt{94+42\sqrt{5}}\right)\left(94-2.21\sqrt{5}\right)}\)
\(A^2=188-2\sqrt{\left(94+2.21\sqrt{5}\right)\left(94-2.21\sqrt{5}\right)}\)
\(A^2=188-2\sqrt{94^2-\left(42\sqrt{5}\right)^2}\)
\(A^2=188-2\sqrt{16}\)
\(A^2=188-8\)
\(A^2=180\)
\(A=6\sqrt{5}\)
\(A=\sqrt{x+2\sqrt{2x-4}}+\sqrt{x-2\sqrt{2x-4}}\left(ĐK:x\ge2\right)\)
\(=\sqrt{2+2\sqrt{2}.\sqrt{x-2}+x-2}+\sqrt{2-2\sqrt{2}.\sqrt{x-2}+x-2}\)
\(=\sqrt{\left(\sqrt{2}+\sqrt{x-2}\right)^2}+\sqrt{\left(\sqrt{2}-\sqrt{x-2}\right)^2}\)
\(=|\sqrt{2}+\sqrt{x-2}|+|\sqrt{2}-\sqrt{x-2}|\)
\(TH1:\sqrt{2}-\sqrt{x-2}\ge0\Rightarrow\sqrt{x-2}\le\sqrt{2}\Leftrightarrow x-2\le2\Rightarrow x\le4\)
Với \(2\le x\le4\)thì \(|\sqrt{2}-\sqrt{x-2}|=\sqrt{2}-\sqrt{x-2}\)
Ta có: \(\sqrt{2}+\sqrt{x-2}+\sqrt{2}-\sqrt{x-2}=2\sqrt{2}\)
\(TH2:\sqrt{2}-\sqrt{x-2}< 0\Leftrightarrow\sqrt{x-2}>\sqrt{2}\Leftrightarrow x-2>2\Leftrightarrow x>4\)
Với \(x>4\)thì \(|\sqrt{2}-\sqrt{x-2}|=\sqrt{x-2}-\sqrt{2}\)
Ta có: \(\sqrt{2}+\sqrt{x-2}-\sqrt{2}+\sqrt{x-2}=2\sqrt{x-2}\)
