Question

Rút gọn:

a. \(\sqrt{13+30\sqrt{2+\sqrt{9+4\sqrt{2}}}}\)

b. \(\sqrt{m+2\sqrt{m-1}}+\sqrt{m-2\sqrt[]{m-1}}\)

\(c.\sqrt{2+\sqrt{3}}.\sqrt{2+\sqrt{2+\sqrt{3}}.}\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}.}\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{ }}3}}\)

Answer 1

Answer 2

\(\text{a) }\sqrt{13+30\sqrt{2+\sqrt{9+4\sqrt{2}}}}\\ =\sqrt{13+30\sqrt{2+\sqrt{8+1+4\sqrt{2}}}}\\ =\sqrt{13+30\sqrt{2+\sqrt{\left(\sqrt{8}+1\right)^2}}}\\ =\sqrt{13+30\sqrt{2+\sqrt{8}+1}}\\ =\sqrt{13+30\sqrt{\left(\sqrt{2}+1\right)^2}}\\ =\sqrt{13+30\sqrt{2}+30}\\ =\sqrt{43+30\sqrt{2}}\\ =\sqrt{25+18+30\sqrt{2}}\\ =\sqrt{\left(5+\sqrt{18}\right)^2}\\ =5+3\sqrt{2}\)

\(\text{b) }\sqrt{m+2\sqrt{m-1}}+\sqrt{m-2\sqrt{m-1}}\\ =\sqrt{m-1+2\sqrt{m-1}+1}+\sqrt{m-1-2\sqrt{m-1}+1}\\ =\sqrt{\left(\sqrt{m-1}+1\right)^2}+\sqrt{\left(\sqrt{m-1}-1\right)^2}\\ =\sqrt{m-1}+1+\sqrt{m-1}-1\\ =2\sqrt{m-1}\)