Lời giải:
\(\sqrt{3+\sqrt{5}}+\sqrt{3-\sqrt{5}}=\sqrt{\frac{6+2\sqrt{5}}{2}}+\sqrt{\frac{6-2\sqrt{5}}{2}}\)
\(\sqrt{\frac{5+1+2\sqrt{5}}{2}}+\sqrt{\frac{5+1-2\sqrt{5}}{2}}=\sqrt{\frac{(\sqrt{5}+1)^2}{2}}+\sqrt{\frac{(\sqrt{5}-1)^2}{2}}\)
\(=\frac{\sqrt{5}+1}{\sqrt{2}}+\frac{\sqrt{5}-1}{\sqrt{2}}=\sqrt{10}\)
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\(\sqrt{8-2\sqrt{15}}-\sqrt{23-4\sqrt{15}}=\sqrt{3+5-2\sqrt{3.5}}-\sqrt{20+3-2\sqrt{20.3}}\)
\(=\sqrt{(\sqrt{5}-\sqrt{3})^2}-\sqrt{(\sqrt{20}-\sqrt{3})^2}\)
\(=\sqrt{5}-\sqrt{3}-(\sqrt{20}-\sqrt{3})=\sqrt{5}-\sqrt{20}=\sqrt{5}-2\sqrt{5}=-\sqrt{5}\)
\(9+4\sqrt{2}=9+2\sqrt{8}=8+1+2\sqrt{8}=(\sqrt{8}+1)^2\)
\(\Rightarrow \sqrt{9+4\sqrt{2}}=\sqrt{8}+1=2\sqrt{2}+1\)
\(\Rightarrow 2+\sqrt{9+4\sqrt{2}}=3+2\sqrt{2}=2+1+2\sqrt{2}=(\sqrt{2}+1)^2\)
\(\Rightarrow \sqrt{2+\sqrt{9+4\sqrt{2}}}=\sqrt{2}+1\)
\(\Rightarrow 13+30\sqrt{2+\sqrt{9+4\sqrt{2}}}=13+30(\sqrt{2}+1)=43+30\sqrt{2}\)
\(\Rightarrow \sqrt{13+30\sqrt{2+\sqrt{9+4\sqrt{2}}}}=\sqrt{43+30\sqrt{2}}\)
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\((\sqrt{3}-\sqrt{2})\sqrt{5+2\sqrt{6}}=(\sqrt{3}-\sqrt{2})\sqrt{2+3+2\sqrt{2.3}}\)
\(=(\sqrt{3}-\sqrt{2})\sqrt{(\sqrt{2}+\sqrt{3})^2}\)
\(=(\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2})=3-2=1\)
