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\(\frac{A}{\sqrt{2}}=\frac{1+\sqrt{7}}{2+\sqrt{8+2\sqrt{7}}}+\frac{1-\sqrt{7}}{2-\sqrt{8-2\sqrt{7}}}\)
\(=\frac{1+\sqrt{7}}{2+1+\sqrt{7}}+\frac{1-\sqrt{7}}{2-\sqrt{7}+1}\)
\(=\frac{1+\sqrt{7}}{3+\sqrt{7}}+\frac{1-\sqrt{7}}{3-\sqrt{7}}\)
=\(\frac{\left(1+\sqrt{7}\right)\left(3-\sqrt{7}\right)+\left(1-\sqrt{7}\right)\left(3+\sqrt{7}\right)}{\left(3+\sqrt{7}\right)\left(3-\sqrt{7}\right)}\)
\(=\frac{-8}{2}=-4\)
\(\Rightarrow A=-4\sqrt{2}\)
a)Bình phương 2 vế ta đc
\(A^2=\left(\sqrt{4}+\sqrt{7}+\sqrt{4}-\sqrt{7}\right)^2\)
⇔ \(A^2=4+\sqrt{7}+2\sqrt{\left(4+\sqrt{7}\right)\left(4-\sqrt{7}\right)}+4-\sqrt{7}\)
⇔ \(A^2=8+2\sqrt{16-7}=8+6=14\)
Vì A luôn ≥ 0 => A = \(\sqrt{14}\)
b) B = \(\frac{\sqrt{2.2}+\sqrt{2.3}+\sqrt{2.5}+\sqrt{2}+\sqrt{3}+\sqrt{5}}{2\sqrt{2}+2\sqrt{3}+2\sqrt{5}}\) . \(\frac{\sqrt{2}-1}{3}\)
= \(\frac{\left(\sqrt{2}+1\right)\left(\sqrt{2}+\sqrt{3}+\sqrt{5}\right)}{2\left(\sqrt{2}+\sqrt{3}+\sqrt{5}\right)}\). \(\frac{\sqrt{2}-1}{3}\)
= \(\frac{\left(\sqrt{2}+1\right)\left(\sqrt{2}-1\right)}{2.3}\)
= \(\frac{1}{6}\)
\(\frac{2}{\sqrt{7}-5}-\frac{2}{\sqrt{7}+5}=\frac{2\sqrt{7}+10}{\left(\sqrt{7}-5\right)\left(\sqrt{7}+5\right)}-\frac{2\sqrt{7}-10}{\left(\sqrt{7}-5\right)\left(\sqrt{7}+5\right)}=\frac{20}{7-25}=\frac{20}{-18}=\frac{10}{-9}\)
\(\frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}-\sqrt{5}}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}=\frac{\left(\sqrt{7}+\sqrt{5}\right)^2+\left(\sqrt{7}-\sqrt{5}\right)^2}{\left(\sqrt{7}-\sqrt{5}\right)\left(\sqrt{7}+\sqrt{5}\right)}=\frac{12+2\sqrt{35}+12-2\sqrt{35}}{2}=\frac{24}{2}=12\)
\(\left(\frac{\sqrt{14}-\sqrt{7}}{1-\sqrt{2}}+\frac{\sqrt{15}-\sqrt{5}}{1-\sqrt{3}}\right)\frac{1}{\sqrt{7}-\sqrt{5}}=\left(\frac{-\sqrt{7}\left(1-\sqrt{2}\right)}{1-\sqrt{2}}+\frac{-\sqrt{5}\left(1-\sqrt{3}\right)}{1-\sqrt{3}}\right)\frac{1}{\sqrt{7}-\sqrt{5}}=\frac{\left(\sqrt{7}+\sqrt{5}\right)}{\sqrt{5}-\sqrt{7}}=\frac{\left(\sqrt{7}+\sqrt{5}\right)^2}{\left(\sqrt{5}-\sqrt{7}\right)\left(\sqrt{5}+\sqrt{7}\right)}=\frac{12+2\sqrt{35}}{-2}=-6-\sqrt{35}\)
\(\frac{3}{\sqrt{5}-2}+\frac{2}{\sqrt{5}+3}-\frac{1}{\sqrt{5}+4}=\frac{3\left(\sqrt{5}+2\right)}{5-4}+\frac{2\left(\sqrt{5}-3\right)}{5-9}-\frac{\sqrt{5}-4}{5-16}\)
\(=3\sqrt{5}+6+\frac{2\sqrt{5}-6}{-4}+\frac{4-\sqrt{5}}{-11}=\frac{66\sqrt{5}+132}{22}+\frac{33-11\sqrt{5}}{22}+\frac{2\sqrt{5}-8}{22}\)
\(=\frac{66\sqrt{5}-11\sqrt{5}+2\sqrt{5}+132+33-8}{22}=\frac{57\sqrt{5}+157}{22}\)
\(A=\frac{\sqrt{3-\sqrt{7}}-\sqrt{3+\sqrt{7}}}{\sqrt{3-\sqrt{2}}}\)
\(\Rightarrow A^2=\frac{3-\sqrt{7}+3+\sqrt{7}-2\sqrt{\left(3-\sqrt{7}\right)\left(3+\sqrt{7}\right)}}{3-\sqrt{2}}\)
\(=\frac{6-2\sqrt{3^2-7}}{3-\sqrt{2}}\)\(=\frac{6-2\sqrt{2}}{3-\sqrt{2}}=\frac{2\left(3-\sqrt{2}\right)}{3-\sqrt{2}}=2\)
Hay \(A^2=2\Rightarrow\orbr{\begin{cases}A=\sqrt{2}\\A=-\sqrt{2}\end{cases}}\)