Question

căn( 2 + căn 3) - căn(2- căn 3)

Answer 1

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

\(=\dfrac{\sqrt{2\left(2+\sqrt{3}\right)}}{\sqrt{2}}-\dfrac{\sqrt{2\left(2-\sqrt{3}\right)}}{\sqrt{2}}=\dfrac{\sqrt{4+2\sqrt{3}}-\sqrt{4-2\sqrt{3}}}{\sqrt{2}}\)

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

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