Question

[prayforme]

Rút gọn

A= \(\sqrt{4+\sqrt{10+2\sqrt{5}}}+\sqrt{4-\sqrt{10+2\sqrt{5}}}\)

B= \(\sqrt{4+\sqrt{15}}+\sqrt{4-\sqrt{15}}-2\sqrt{3-5}\)

Answer 1

[Phương An]

\(M=\sqrt{4+\sqrt{10+2\sqrt{5}}}+\sqrt{4-\sqrt{10+2\sqrt{5}}}\)

\(\Rightarrow M^2=\left(4+\sqrt{10+2\sqrt{5}}\right)+2\sqrt{\left(\sqrt{4+\sqrt{10+2\sqrt{5}}}\right)\left(\sqrt{4-\sqrt{10+2\sqrt{5}}}\right)}+\left(4-\sqrt{10+2\sqrt{5}}\right)\)

\(=8+2\sqrt{16-\left(10+2\sqrt{5}\right)}\)

\(=8+2\sqrt{\left(\sqrt{5}-1\right)^2}\)

\(=8+2\sqrt{5}-2\)

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

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

\(\Rightarrow\left|M\right|=\sqrt{5}+1\) mà M > 0

\(\Rightarrow M=\sqrt{5}+1\)

Answer 2

[Phương An]

\(M=\sqrt{4+\sqrt{15}}+\sqrt{4-\sqrt{15}}-2\sqrt{3-\sqrt{5}}\)

\(\Rightarrow\sqrt{2}M=\sqrt{8+2\sqrt{15}}+\sqrt{8-2\sqrt{15}}-2\sqrt{6-2\sqrt{5}}\)

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

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

= 2

\(\Rightarrow M=\sqrt{2}\)