Bài 1. Thực hiện phép tính
a, \(\left(\sqrt{3}+1\right)^2+\left(1-\sqrt{3}\right)^2\)
b, \(\left(\sqrt{28}-2\sqrt{3}+\sqrt{7}\right)\sqrt{7}+\sqrt{84}\)
Bài 2. Trục căn thức ở mẫu
a, \(\frac{1}{1+\sqrt{2}+\sqrt{3}}\)
b, \(\frac{3\sqrt{2}}{\sqrt{3}+1}\)
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b)\(\frac{3\sqrt{2}}{\sqrt{3}+1}\)
\(=\frac{3\sqrt{2}\left(\sqrt{3}-1\right)}{(\sqrt{3}+1)\left(\sqrt{3}-1\right)}\)
\(=\frac{3\left(\sqrt{6}-\sqrt{2}\right)}{3-1}\)
\(=\frac{3\left(\sqrt{6}-\sqrt{2}\right)}{2}\)
a)\(\left(\sqrt{3}+1\right)^2+\left(1-\sqrt{3}\right)^2\)
\(=3+2\sqrt{3}+1+1-2\sqrt{3}+3\)
\(=8\)
b)\(\left(\sqrt{28}-2\sqrt{3}+\sqrt{7}\right)\sqrt{7}+\sqrt{84}\)
\(=\sqrt{28.7}-2\sqrt{3.7}+\sqrt{7}.\sqrt{7}+\sqrt{84}\)
\(=\sqrt{196}-2\sqrt{21}+7+\sqrt{4.21}\)
\(=\sqrt{14^2}-2\sqrt{21}+7+2\sqrt{21}\)
\(=14-2\sqrt{21}+7+2\sqrt{21}\)
\(=21\)
a)\(\frac{1}{1+\sqrt{2}+\sqrt{3}}\)
\(=\frac{1\left(1+\sqrt{2}-\sqrt{3}\right)}{(1+\sqrt{2}+\sqrt{3})\left(1+\sqrt{2}-\sqrt{3}\right)}\)
\(=\frac{1+\sqrt{2}-\sqrt{3}}{1+\sqrt{2}-\sqrt{3}+\sqrt{2}+2-\sqrt{6}+\sqrt{3}+\sqrt{6}-3}\)
\(=\frac{1+\sqrt{2}-\sqrt{3}}{2\sqrt{2}}\)
\(=\frac{\sqrt{2}.(1+\sqrt{2}-\sqrt{3})}{2\sqrt{2}\cdot\sqrt{2}}\)
\(=\frac{\sqrt{2}.(1+\sqrt{2}-\sqrt{3})}{4}\)
