\(A=\left(\sqrt{5}-1\right)\sqrt{6+2\sqrt{5}}=\left(\sqrt{5}-1\right)\sqrt{5+2\sqrt{5}+1}=\left(\sqrt{5}-1\right)\sqrt{\left(\sqrt{5}+1\right)^2}=\)

\(=\left(\sqrt{5}-1\right)\left(\sqrt{5}+1\right)=\left(\sqrt{5}\right)^2-1=4\).

A = 4

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

=\(5+\frac{2\sqrt{6}}{\sqrt{3}}+\sqrt{2}-5+\frac{2\sqrt{6}}{\sqrt{3}}+\sqrt{2}\)

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

=\(0+\frac{4\sqrt{6}}{\sqrt{3}}+2\sqrt{2}\)

=\(\frac{4\sqrt{2}.\sqrt{3}}{\sqrt{3}}+2\sqrt{2}\)

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

=\(6\sqrt{2}\)

\(-\sqrt{5-2\sqrt{6}}=-\sqrt{\left(\sqrt{3}-\sqrt{2}\right)^2}=-\left|\sqrt{3}-\sqrt{2}\right|=\sqrt{2}-\sqrt{3}\)

\(D=\sqrt{\frac{\left(5+2\sqrt{6}\right)^2}{25-24}}+\sqrt{\frac{\left(5-2\sqrt{6}\right)^2}{25-24}}=5+2\sqrt{6}+5-2\sqrt{6}=10\)

\(\sqrt{\left(6+2\sqrt{5}\right)^3}-\sqrt{\left(6-2\sqrt{5}\right)^3}\)

\(=\sqrt{\left(6+2\sqrt{5}\right)^2.\left(6+2\sqrt{5}\right)}+\sqrt{\left(6-2\sqrt{5}\right)^2.\left(6-2\sqrt{5}\right)}\)

\(=\left(6+2\sqrt{5}\right).\sqrt{\sqrt{5}^2+2\sqrt{5}+\sqrt{1}^2}+\left(6-2\sqrt{5}\right).\sqrt{\sqrt{5}^2-2\sqrt{5}+\sqrt{1}^2}\)

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

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

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

bạn có thể phân tích tiếp bằng hđt

Bài làm:

Xét: \(\sqrt{2}< \sqrt{11+6\sqrt{2}}\)

=> \(\sqrt{2}-\sqrt{11+6\sqrt{2}}< 0\) (1)

và \(5>\sqrt{5}\) => \(5-\sqrt{5}>0\)

<=> \(2\sqrt{5-\sqrt{5}}>0\) => \(\sqrt{6+2\sqrt{5-\sqrt{5}}}>0\) (2)

Từ (1) và (2)

=> \(\frac{\sqrt{2}-\sqrt{11+6\sqrt{2}}}{\sqrt{6+2\sqrt{5-\sqrt{5}}}}< 0\)

Mà biểu thức trong căn phải có giá trị không âm

=> Mâu thuẫn

=> Căn thức không có giá trị