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a. \(\left(\sqrt{28}-2\sqrt{14}+\sqrt{7}\right)\sqrt{7}+7\sqrt{8}=\left(2\sqrt{7}-2\sqrt{14}+\sqrt{7}\right)\sqrt{7}+14\sqrt{2}=14-14\sqrt{2}+7+14\sqrt{2}=21\)
b. \(\dfrac{\sqrt{15}-\sqrt{5}}{\sqrt{3}-1}-\dfrac{5-2\sqrt{5}}{2\sqrt{5}-4}=\dfrac{\sqrt{5}\left(\sqrt{3}-1\right)}{\sqrt{3}-1}-\dfrac{\sqrt{5}\left(\sqrt{5}-2\right)}{2\left(\sqrt{5}-2\right)}=\sqrt{5}-\dfrac{\sqrt{5}}{2}=\dfrac{2\sqrt{5}-\sqrt{5}}{2}=\dfrac{\sqrt{5}}{2}\)
c. \(\dfrac{\sqrt{6}+\sqrt{14}}{2\sqrt{3}+\sqrt{28}}=\dfrac{\sqrt{6}+\sqrt{14}}{2\sqrt{3}+2\sqrt{7}}=\dfrac{\sqrt{2}\left(\sqrt{3}+\sqrt{7}\right)}{2\left(\sqrt{3}+\sqrt{7}\right)}=\dfrac{\sqrt{2}}{2}\)
\(1.\dfrac{6}{1-\sqrt{3}}-\dfrac{3\sqrt{3}+3}{\sqrt{3}+1}=\dfrac{6}{1-\sqrt{3}}-\dfrac{3\left(\sqrt{3}+1\right)}{\sqrt{3}+1}=\dfrac{6}{1-\sqrt{3}}-3=\dfrac{3+3\sqrt{3}}{1-\sqrt{3}}\) \(2.\dfrac{\sqrt{12}-6}{\sqrt{8}-\sqrt{24}}-\dfrac{3+\sqrt{3}}{\sqrt{3}}+\dfrac{4}{1-\sqrt{7}}=\dfrac{2\sqrt{3}-6}{2\sqrt{2}-2\sqrt{6}}-\dfrac{\sqrt{3}\left(\sqrt{3}+1\right)}{\sqrt{3}}=\dfrac{2\sqrt{3}\left(1-\sqrt{3}\right)}{2\sqrt{2}\left(1-\sqrt{3}\right)}-\sqrt{3}-1=\dfrac{\sqrt{3}}{\sqrt{2}}-\sqrt{3}-1=\dfrac{\sqrt{3}-\sqrt{6}-\sqrt{2}}{\sqrt{2}}\) \(3.\left(\dfrac{\sqrt{14}-\sqrt{7}}{1-\sqrt{2}}+\dfrac{\sqrt{15}-\sqrt{5}}{1-\sqrt{3}}\right):\dfrac{1}{\sqrt{7}-\sqrt{5}}=\left[\dfrac{-\sqrt{7}\left(1-\sqrt{2}\right)}{1-\sqrt{2}}-\dfrac{\sqrt{5}\left(1-\sqrt{3}\right)}{1-\sqrt{3}}\right].\left(\sqrt{7}-\sqrt{5}\right)=-\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)=-2\) \(4.\dfrac{\left(\sqrt{2}+1\right)^2-4\sqrt{2}}{\sqrt{2}-1}.\left(\sqrt{2}+1\right)=\dfrac{\left(2-2\sqrt{2}+1\right)\left(\sqrt{2}+1\right)}{\sqrt{2}-1}=\dfrac{\left(\sqrt{2}-1\right)^2\left(\sqrt{2}+1\right)}{\sqrt{2}-1}=1\)
\(a.\dfrac{2}{4-3\sqrt{2}}-\dfrac{2}{4+3\sqrt{2}}=\dfrac{2\left(4+3\sqrt{2}\right)-2\left(4-3\sqrt{2}\right)}{\left(4-3\sqrt{2}\right)\left(4+3\sqrt{2}\right)}=\dfrac{8+6\sqrt{2}-8+6\sqrt{2}}{16-18}=\dfrac{12\sqrt{2}}{-2}=-6\sqrt{2}\)\(b.\dfrac{2}{1+\sqrt{2}}+\dfrac{2}{1-\sqrt{2}}=\dfrac{2\left(1-\sqrt{2}\right)+2\left(1+\sqrt{2}\right)}{\left(1+\sqrt{2}\right)\left(1-\sqrt{2}\right)}=\dfrac{2-2\sqrt{2}+2+2\sqrt{2}}{1-2}=-4\)\(c.\left(\sqrt{14}-3\sqrt{2}\right)^2+6\sqrt{28}=14-12\sqrt{7}+18+12\sqrt{7}=14+18=32\)\(d.\left(\sqrt{28}-2\sqrt{14}+\sqrt{7}\right).\sqrt{7}+7\sqrt{8}=14-14\sqrt{2}+7+14\sqrt{2}=14+7=21\)\(e.\left(\sqrt{6}-\sqrt{5}\right)^2-2\sqrt{120}=6-2\sqrt{30}+5-4\sqrt{30}=11-6\sqrt{30}\)
a: \(=\left(-\sqrt{5}-\sqrt{7}\right)\cdot\left(\sqrt{7}-\sqrt{5}\right)\)
\(=-\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)\)
=-2
b: \(=\sqrt{2-\sqrt{3}}+\sqrt{2+\sqrt{3}}\)
\(=\dfrac{\sqrt{4-2\sqrt{3}}+\sqrt{4+2\sqrt{3}}}{\sqrt{2}}\)
\(=\dfrac{\sqrt{3}-1+\sqrt{3}+1}{\sqrt{2}}=\sqrt{6}\)
c: \(=\dfrac{\sqrt{10}\left(\sqrt{2}-\sqrt{5}\right)}{\sqrt{2}-\sqrt{5}}-2-\sqrt{10}+3\sqrt{7}+2\)
\(=\sqrt{10}-\sqrt{10}+3\sqrt{7}=3\sqrt{7}\)
Lời giải:
\(\left(\frac{\sqrt{14}-\sqrt{7}}{1-\sqrt{2}}+\frac{\sqrt{15}-\sqrt{5}}{1-\sqrt{3}}\right): \frac{1}{\sqrt{7}-\sqrt{5}}\)
\(=\left(\frac{\sqrt{7}(\sqrt{2}-1)}{1-\sqrt{2}}+\frac{\sqrt{5}(\sqrt{3}-1)}{1-\sqrt{3}}\right).(\sqrt{7}-\sqrt{5})\)
\(=(-\sqrt{7}-\sqrt{5}). (\sqrt{7}-\sqrt{5})=-(\sqrt{7}+\sqrt{5})(\sqrt{7}-\sqrt{5})\)
\(=-[(\sqrt{7})^2-(\sqrt{5})^2]=-2\)
a: \(=\left(2\sqrt{7}+\sqrt{7}+2\sqrt{14}\right)\cdot\sqrt{7}-\left(51+14\sqrt{2}\right)\)
\(=3\sqrt{7}\cdot\sqrt{7}+2\sqrt{14}\cdot\sqrt{7}-51-14\sqrt{2}\)
\(=21-51=-30\)
b: \(=\dfrac{\sqrt{10}}{2}+\dfrac{\sqrt{10}-\sqrt{6}}{2}=\dfrac{2\sqrt{10}-\sqrt{6}}{2}\)
c: \(=\dfrac{\left(\sqrt{5}+\sqrt{3}\right)^2}{\sqrt{5}+\sqrt{3}}+\dfrac{\left(\sqrt{5}-\sqrt{2}\right)^2}{\sqrt{5}-\sqrt{2}}\)
\(=\sqrt{5}+\sqrt{3}+\sqrt{5}-\sqrt{2}\)
\(=2\sqrt{5}+\sqrt{3}-\sqrt{2}\)
\(\sqrt{112}-7\sqrt{\dfrac{1}{7}}-14\sqrt{\dfrac{1}{28}}-\dfrac{21}{\sqrt{7}}=\sqrt{16.7}-\sqrt{49.\dfrac{1}{7}}-2.\sqrt{\dfrac{1}{4}.49.\dfrac{1}{7}}-\dfrac{3.7}{\sqrt{7}}\)
\(=4\sqrt{7}-\sqrt{7}-2.\dfrac{1}{2}\sqrt{7}-3\sqrt{7}=4\sqrt{7}-\sqrt{7}-\sqrt{7}-3\sqrt{7}=-\sqrt{7}\)
\(=\sqrt{4^2.7}-\sqrt{\dfrac{7^2}{7}}-\sqrt{\dfrac{14^2}{\text{28}}}-\sqrt{3^2.7}\)
\(=4\sqrt{7}-\sqrt{7}-\sqrt{7}-3\sqrt{7}\)
\(=\sqrt{7}\left(4-1-1-3\right)\)
\(=-\sqrt{7}\)