Question

1.\(\sqrt{3+\sqrt{5}}\left(\sqrt{3-\sqrt[]{5}}-2\sqrt{2}\right)\)

2.\(\sqrt{13+4\sqrt{3}}-\sqrt{\left(3-2\sqrt{3}\right)^2}\)

Answer 1

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

2.\(\sqrt{13+4\sqrt{3}}-\sqrt{\left(3-2\sqrt{3}\right)^2}=\sqrt{12+2.2\sqrt{3}+1}-\left|3-2\sqrt{3}\right|=\sqrt{\left(2\sqrt{3}+1\right)^2}-\left(2\sqrt{3}-3\right)=2\sqrt{3}+1-2\sqrt{3}+3=4\)