Question

[Wang Karry]

Rút gọn các biểu thức sau:

A =\(\sqrt{5}-\sqrt{3-\sqrt{29-12\sqrt{5}}}\)

B = \(\sqrt{3-\sqrt{5}}.\left(\sqrt{10}-\sqrt{2}\right)\left(3+\sqrt{5}\right)\)

Answer 1

[Phương An]

\(\sqrt{5}-\sqrt{3-\sqrt{29-12\sqrt{5}}}\)

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

\(=\sqrt{5}-\sqrt{6-2\sqrt{5}}\)

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

= 1

\(\sqrt{3-\sqrt{5}}\left(\sqrt{10}-\sqrt{2}\right)\left(3+\sqrt{5}\right)\)

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

\(=\sqrt{9-5}\times\sqrt{6+2\sqrt{5}}\left(\sqrt{5}-1\right)\)

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

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

= 8

Answer 2

[qwerty]

a) \(A=\sqrt{5}-\sqrt{3-\sqrt{29-12\sqrt{5}}}\)

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

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

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

\(=\sqrt{5}-\sqrt{6-2\sqrt{5}}\)

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

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

\(=\sqrt{5}-\sqrt{5}+1\)

\(=1\)

b) \(B=\sqrt{3-\sqrt{5}}\cdot\left(\sqrt{10}-\sqrt{2}\right)\left(3+\sqrt{5}\right)\)

\(=\sqrt{3-\sqrt{5}}\left(3\sqrt{10}+\sqrt{50}-3\sqrt{2}-\sqrt{10}\right)\)

\(=\sqrt{3-\sqrt{5}}\left(3\sqrt{10}+5\sqrt{2}-3\sqrt{2}-\sqrt{10}\right)\)

\(=\sqrt{3-\sqrt{5}}\left(2\sqrt{10}+2\sqrt{2}\right)\)

\(=\sqrt{3-\sqrt{5}}\sqrt{\left(2\sqrt{10}+2\sqrt{2}\right)^2}\)

\(=\sqrt{\left(3-\sqrt{5}\right)\left(2\sqrt{10}+2\sqrt{2}\right)^2}\)

\(=\sqrt{\left(3-\sqrt{5}\right)\left(4\cdot10+8\sqrt{20}+4\cdot2\right)}\)

\(=\sqrt{\left(3-\sqrt{5}\right)\left(40+16\sqrt{5}+8\right)}\)

\(=\sqrt{\left(3-\sqrt{5}\right)\left(48+16\sqrt{5}\right)}\)

\(=\sqrt{\left(3-\sqrt{5}\right)\cdot16\left(3+\sqrt{5}\right)}\)

\(=\sqrt{\left(9-5\right)\cdot16}\)

\(=\sqrt{4\cdot16}\)

\(=\sqrt{64}\)

\(=8\)