\(\sqrt{49-5\sqrt{96}}+\sqrt{49+5\sqrt{96}}=\sqrt{25-2\cdot5\cdot2\sqrt{6}+24}+\sqrt{25-2\cdot5\cdot2\sqrt{6}+24}=\sqrt{\left(5+2\sqrt{6}\right)^2}+\sqrt{\left(5-2\sqrt{6}\right)^2}=5+2\sqrt{6}+5-2\sqrt{6}=10\) ---
\(\sqrt{13-\sqrt{160}}+\sqrt{53+4\sqrt{90}}=\sqrt{8-2\sqrt{5}\cdot\sqrt{8}+5}+\sqrt{45+2\cdot3\sqrt{5}\cdot\sqrt{8}+8}=\sqrt{\left(\sqrt{8}-\sqrt{5}\right)^2}+\sqrt{\left(3\sqrt{5}+\sqrt{8}\right)^2}=\sqrt{8}-\sqrt{5}+3\sqrt{5}+\sqrt{8}=2\sqrt{8}+2\sqrt{5}\)
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\(\sqrt{11-6\sqrt{2}}+\sqrt{3-2\sqrt{2}}=\sqrt{9-2\cdot3\cdot\sqrt{2}+2}+\sqrt{2-2\sqrt{2}+1}=\sqrt{\left(3-\sqrt{2}\right)^2}+\sqrt{\left(\sqrt{2}-1\right)^2}=3-\sqrt{2}+\sqrt{2}-1=2\)
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\(\sqrt{15-6\sqrt{6}}+\sqrt{35-12\sqrt{6}}=\sqrt{9-2\cdot3\cdot\sqrt{6}+6}+\sqrt{27-2\cdot\sqrt{27}\cdot\sqrt{8}+8}=\sqrt{\left(3-\sqrt{6}\right)^2}+\sqrt{\left(3\sqrt{3}-2\sqrt{2}\right)^2}=3-\sqrt{6}+3\sqrt{3}-2\sqrt{2}\)
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\(\sqrt{17-3\sqrt{32}}+\sqrt{17+3\sqrt{32}}=\sqrt{9-2\cdot3\cdot2\sqrt{2}+8}+\sqrt{9+2\cdot2\cdot2\sqrt{2}+8}=\sqrt{\left(3-2\sqrt{2}\right)^2}+\sqrt{\left(3+2\sqrt{2}\right)^2}=3-2\sqrt{2}+3+2\sqrt{2}=6\)
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\(\sqrt{49-5\sqrt{96}}+\sqrt{49+5\sqrt{96}}\)
\(=\sqrt{25-2\times5\sqrt{24}+24}+\sqrt{25+2\times5\sqrt{24}+24}\)
\(=\sqrt{\left(5-\sqrt{24}\right)^2}+\sqrt{\left(5+\sqrt{24}\right)^2}\)
\(=5-\sqrt{24}+5+\sqrt{24}\)
\(=10\)
\(\sqrt{11-6\sqrt{2}}+\sqrt{3-2\sqrt{2}}\)
\(=\sqrt{9-2\times3\sqrt{2}+2}+\sqrt{2-2\times1\sqrt{2}+1}\)
\(=\sqrt{\left(3-\sqrt{2}\right)^2}+\sqrt{\left(\sqrt{2}-1\right)^2}\)
\(=3-\sqrt{2}+\sqrt{2}-1\)
\(=2\)
\(\sqrt{15-6\sqrt{6}}+\sqrt{35-12\sqrt{6}}\)
\(=\sqrt{9-2\times3\sqrt{6}+6}+\sqrt{9-2\times3\sqrt{24}+24}\)
\(=\sqrt{\left(3-\sqrt{6}\right)^2}+\sqrt{\left(3-\sqrt{24}\right)^2}\)
\(=3-\sqrt{6}+\sqrt{24}-3\)
\(=\sqrt{6}\)
