Question

\(\left(\sqrt{10}-\sqrt{2}\right)\left(3+\sqrt{5}\right)\sqrt{3-\sqrt{5}}\)

Answer 1

\(\left(\sqrt{10}-\sqrt{2}\right)\left(3+\sqrt{5}\right)\sqrt{3-\sqrt{5}}=\left(\sqrt{10}-\sqrt{2}\right)\sqrt{\left(3+\sqrt{5}\right)^2\left(3-\sqrt{5}\right)}=\left(\sqrt{10}-\sqrt{2}\right)\sqrt{\left(3+\sqrt{5}\right)\left(3-\sqrt{5}\right)\left(3+\sqrt{5}\right)}=\sqrt{2}\left(\sqrt{5}-1\right)\sqrt{\left(9-5\right)\left(3+\sqrt{5}\right)}=\sqrt{2}\left(\sqrt{5}-1\right)\sqrt{4\left(3+\sqrt{5}\right)}=\sqrt{2}\left(\sqrt{5}-1\right).2\sqrt{3+\sqrt{5}}=2\left(\sqrt{5}-1\right)\sqrt{6+2\sqrt{5}}=2\left(\sqrt{5}-1\right)\sqrt{5+2\sqrt{5}+1}=2\left(\sqrt{5}-1\right)\sqrt{\left(\sqrt{5}+1\right)^2}=2\left(\sqrt{5}-1\right)\left(\sqrt{5}+1\right)=2\left(5-1\right)=2.4=8\)