Question

Rút gọn các biểu thức sau

a)

\(\left(2\sqrt{4+\sqrt{6-2\sqrt{5}}}\right)\left(\sqrt{10}-\sqrt{2}\right)\)

b) \(\sqrt{13-\sqrt{160}}-\sqrt{53+4\sqrt{90}}\)

Answer 1

\(\left(2\sqrt{4+\sqrt{6-2\sqrt{5}}}\right)\left(\sqrt{10}-\sqrt{2}\right)\)

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

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

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

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

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

\(=2\left(5-1\right)\)

= 8

~ ~ ~

\(\sqrt{13-\sqrt{160}}-\sqrt{53+4\sqrt{90}}\)

\(=\sqrt{13-4\sqrt{10}}-\sqrt{53+12\sqrt{10}}\)

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

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

\(=-4\sqrt{5}\)

Answer 2

a. \(\left(2\sqrt{4+\sqrt{6-2\sqrt{5}}}\right)\left(\sqrt{10}-\sqrt{2}\right)=\left[2\sqrt{4+\sqrt{\left(\sqrt{5}-1\right)^2}}\right]\left(\sqrt{10}-\sqrt{2}\right)=\left(2\sqrt{4+\sqrt{5}-1}\right)\left(\sqrt{10}-\sqrt{2}\right)=\left(2\sqrt{3+\sqrt{5}}\right)\left(\sqrt{10}-\sqrt{2}\right)=\left[2\sqrt{\left(\sqrt{\dfrac{5}{2}}+\sqrt{\dfrac{1}{2}}\right)^2}\right]\left(\sqrt{10}-\sqrt{2}\right)=\left[2\left(\sqrt{\dfrac{5}{2}}+\sqrt{\dfrac{1}{2}}\right)\right]\left(\sqrt{10}-\sqrt{2}\right)=\left(\sqrt{10}+\sqrt{2}\right)\left(\sqrt{10}-\sqrt{2}\right)=10-2=8\)

b. \(\sqrt{13-\sqrt{160}}-\sqrt{53+4\sqrt{90}}=\sqrt{\left(2\sqrt{2}-\sqrt{5}\right)^2}-\sqrt{\left(3\sqrt{5}+2\sqrt{2}\right)^2}=2\sqrt{2}-\sqrt{5}-3\sqrt{5}-2\sqrt{2}=-4\sqrt{5}\)