Question

a)  \(\sqrt{9-4\sqrt{5}}-\sqrt{9+4\sqrt{5}}\)

b)  \(\sqrt{4-\sqrt{7}}-\sqrt{4+\sqrt{7}}\)

c)  \(\sqrt{94-42\sqrt{5}}-\sqrt{94+42\sqrt{5}}\)

Answer 1

a)  \(\sqrt{9-4\sqrt{5}}-\sqrt{9+4\sqrt{5}}=\sqrt{\left(\sqrt{5}-2\right)^2}-\sqrt{\left(\sqrt{5}+2\right)^2}\)

\(=\left(\sqrt{5}-2\right)-\left(\sqrt{5}+2\right)=-4\)

b)   \(\sqrt{4-\sqrt{7}}-\sqrt{4+\sqrt{7}}=\frac{1}{\sqrt{2}}.\left(\sqrt{8-2\sqrt{7}}-\sqrt{8+2\sqrt{7}}\right)\)

\(=\frac{1}{\sqrt{2}}\left(\sqrt{\left(\sqrt{7}-1\right)^2}-\sqrt{\left(\sqrt{7}+1\right)^2}\right)\)

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

c)  \(\sqrt{94-42\sqrt{5}}-\sqrt{94+42\sqrt{5}}=\sqrt{\left(7-3\sqrt{5}\right)^2}-\sqrt{\left(7+3\sqrt{5}\right)^2}\)

\(=7-3\sqrt{5}-\left(7+3\sqrt{5}\right)=-6\sqrt{5}\)