H24
20 tháng 6 2023 lúc 17:07
\(1,\sqrt{13+4\sqrt{3}}+2\sqrt{7-4\sqrt{3}}\)
\(=\sqrt{\sqrt{12^2}+2\sqrt{12}+1}+2\sqrt{\sqrt{3^2}-2.2\sqrt{3}+2^2}\)
\(=\sqrt{\left(\sqrt{12}+1\right)^2}+2\sqrt{\left(\sqrt{3}-2\right)}\)
\(=\left|\sqrt{12}+1\right|+2\left|\sqrt{3}-2\right|\)
\(=\sqrt{12}+1+2\left(2-\sqrt{3}\right)\)
\(=2\sqrt{3}+1+4-2\sqrt{3}\)
\(=5\)
\(2,\left(\sqrt{10}-\sqrt{2}\right).\sqrt{3+\sqrt{5}}\)
\(=\left[\sqrt{2}\left(\sqrt{5}-1\right)\right]\sqrt{3+\sqrt{5}}\)
\(=\sqrt{2\left(3+\sqrt{5}\right)}\left(\sqrt{5}-1\right)\)
\(=\sqrt{6+2\sqrt{5}}\left(\sqrt{5}-1\right)\)
\(=\sqrt{\left(\sqrt{5}+1\right)^2}\left(\sqrt{5}-1\right)\)
\(=\left(\sqrt{5}+1\right)\left(\sqrt{5}-1\right)\)
\(=5-1=4\)
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