Question

Thực hiện phép tính

a) \(\left(4+\sqrt{15}\right)\left(\sqrt{10}-\sqrt{6}\right)\sqrt{4-\sqrt{15}}\)

b) \(\left(3-\sqrt{5}\right)\sqrt{3+\sqrt{5}}+\sqrt{3-\sqrt{5}}\left(3+\sqrt{5}\right)\)

c) \(\sqrt{3+\sqrt{5}}-\sqrt{3-\sqrt{5}}-\sqrt{2}\)

d) \(\sqrt{4-\sqrt{7}}-\sqrt{4+\sqrt{7}}+\sqrt{7}\)

e) \(\sqrt{6,5+\sqrt{12}}+\sqrt{6,5-\sqrt{12}}+2\sqrt{6}\)

Answer 1

\(A=\left(4+\sqrt{15}\right)\left(\sqrt{5}-\sqrt{3}\right)\sqrt{8-2\sqrt{15}}\)

\(=\left(4+\sqrt{15}\right)\left(\sqrt{5}-\sqrt{3}\right)\sqrt{\left(\sqrt{5}-\sqrt{3}\right)^2}\)

\(=\left(4+\sqrt{15}\right)\left(\sqrt{5}-\sqrt{3}\right)^2=\left(4+\sqrt{15}\right)\left(8-2\sqrt{15}\right)\)

\(=2\left(4+\sqrt{15}\right)\left(4-\sqrt{15}\right)=2\left(16-15\right)=2\)

\(B=\frac{1}{\sqrt{2}}\left(\left(3-\sqrt{5}\right)\sqrt{6+2\sqrt{5}}+\sqrt{6-2\sqrt{5}}\left(3+\sqrt{5}\right)\right)\)

\(=\frac{1}{\sqrt{2}}\left(\left(3-\sqrt{5}\right)\sqrt{\left(\sqrt{5}+1\right)^2}+\sqrt{\left(\sqrt{5}-1\right)^2}\left(3+\sqrt{5}\right)\right)\)

\(=\frac{1}{\sqrt{2}}\left(\left(3-\sqrt{5}\right)\left(\sqrt{5}+1\right)+\left(\sqrt{5}-1\right)\left(3+\sqrt{5}\right)\right)\)

\(=\frac{1}{\sqrt{2}}\left(2\sqrt{5}-2+2\sqrt{5}+2\right)=\frac{4\sqrt{5}}{\sqrt{2}}=2\sqrt{10}\)

\(C=\frac{1}{\sqrt{2}}\left(\sqrt{6+2\sqrt{5}}-\sqrt{6-2\sqrt{5}}-2\right)\)

\(=\frac{1}{\sqrt{2}}\left(\sqrt{\left(\sqrt{5}+1\right)^2}-\sqrt{\left(\sqrt{5}-1\right)^2}-2\right)\)

\(=\frac{1}{\sqrt{2}}\left(\sqrt{5}+1-\sqrt{5}+1-2\right)=0\)

Answer 2

\(D=\frac{1}{\sqrt{2}}\left(\sqrt{8-2\sqrt{7}}-\sqrt{8+2\sqrt{7}}+\sqrt{14}\right)\)

\(=\frac{1}{\sqrt{2}}\left(\sqrt{\left(\sqrt{7}-1\right)^2}-\sqrt{\left(\sqrt{7}+1\right)^2}+\sqrt{14}\right)\)

\(=\frac{1}{\sqrt{2}}\left(\sqrt{7}-1-\sqrt{7}-1+\sqrt{14}\right)\)

\(=\frac{1}{\sqrt{2}}\left(-2+\sqrt{14}\right)=\sqrt{7}-\sqrt{2}\)

\(E=\frac{1}{\sqrt{2}}\left(\sqrt{13+2\sqrt{12}}+\sqrt{13-2\sqrt{12}}\right)+2\sqrt{6}\)

\(=\frac{1}{\sqrt{2}}\left(\sqrt{\left(\sqrt{12}+1\right)^2}+\sqrt{\left(\sqrt{12}-1\right)^2}\right)+2\sqrt{6}\)

\(=\frac{1}{\sqrt{2}}\left(\sqrt{12}+1+\sqrt{12}-1\right)+2\sqrt{6}\)

\(=\sqrt{24}+2\sqrt{6}=4\sqrt{6}\)