Question

a) √(4-√7)-√(4+√7) b)√(3+√5)+√(3-√5)

c)√(2+√3)-√(2-√3)

Answer 1

a: \(=\dfrac{1}{\sqrt{2}}\left(\sqrt{8-2\sqrt{7}}-\sqrt{8+2\sqrt{7}}\right)\)

\(=\dfrac{1}{\sqrt{2}}\left(\sqrt{7}-1-\sqrt{7}-1\right)=-2\cdot\dfrac{1}{\sqrt{2}}=-\sqrt{2}\)

b: \(=\dfrac{1}{\sqrt{2}}\left(\sqrt{6+2\sqrt{5}}+\sqrt{6-2\sqrt{5}}\right)\)

\(=\dfrac{1}{\sqrt{2}}\left(\sqrt{5}+1+\sqrt{5}-1\right)=2\sqrt{5}\cdot\dfrac{1}{\sqrt{2}}=\sqrt{10}\)

c: \(=\dfrac{1}{\sqrt{2}}\left(\sqrt{4+2\sqrt{3}}-\sqrt{4-2\sqrt{3}}\right)\)

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