Question

[Quốc Sơn]

Rút gọn căn thức

\(\sqrt{4+\sqrt{15}}-\sqrt{7-3\sqrt{5}}\)

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

Answer 1

[Mashiro Shiina]

\(A=\sqrt{4+\sqrt{15}}-\sqrt{7-3\sqrt{5}}\)

\(\Rightarrow A\sqrt{2}=\sqrt{8+2\sqrt{15}}-\sqrt{14-2\sqrt{45}}\)

\(A\sqrt{2}=\sqrt{\left(\sqrt{3}\right)^2+\left(\sqrt{5}\right)^2+2.\sqrt{3}.\sqrt{5}}-\sqrt{\left(\sqrt{5}\right)^2+\left(\sqrt{9}\right)^2-2.\sqrt{5}.\sqrt{9}}\)

\(A\sqrt{2}=\sqrt{\left(\sqrt{3}+\sqrt{5}\right)^2}-\sqrt{\left(\sqrt{9}-\sqrt{5}\right)^2}\)

\(A\sqrt{2}=\sqrt{3}+\sqrt{5}-\sqrt{9}+\sqrt{5}=2\sqrt{5}+\sqrt{3}-\sqrt{9}\Rightarrow A=\frac{2\sqrt{5}+\sqrt{3}-\sqrt{9}}{\sqrt{2}}\)\(B=\sqrt{2+\sqrt{3}}+\sqrt{2-\sqrt{3}}\)

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

\(B\sqrt{2}=\sqrt{1^2+\left(\sqrt{3}\right)^2+2.1.\sqrt{3}}+\sqrt{\left(\sqrt{3}\right)^2+1^2-2.1.\sqrt{3}}\)\(B\sqrt{2}=\sqrt{\left(1+\sqrt{3}\right)^2}+\sqrt{\left(\sqrt{3}-1\right)^2}=1+\sqrt{3}+\sqrt{3}-1=2\sqrt{3}\)