Thực hiện phép tính:
a)\(\sqrt{13+30\sqrt{2+\sqrt{9+4\sqrt{2}}}}\)
b)\(\left(\sqrt{3}-\sqrt{2}\right)\sqrt{5+2\sqrt{6}}\)
c)\(\sqrt{5-\sqrt{13+4\sqrt{3}}}+\sqrt{3+\sqrt{13+4\sqrt{3}}}\)
d)\(\sqrt{1+\sqrt{3+\sqrt{13+4\sqrt{3}}}}+\sqrt{1-\sqrt{3-\sqrt{13-4\sqrt{3}}}}\)
Mọi người giải gấp giúp em với!!!!
a/ \(\sqrt{13+30\sqrt{2+\sqrt{9+4\sqrt{2}}}}\)
\(=\sqrt{13+30\sqrt{2+\sqrt{1+2\cdot1\cdot2\sqrt{2}+8}}}\)
\(=\sqrt{13+30\sqrt{2+\sqrt{\left(1+2\sqrt{2}\right)^2}}}\)
\(=\sqrt{13+30\sqrt{2+2\sqrt{2}+1}}=\sqrt{13+30\sqrt{\left(\sqrt{2}+1\right)^2}}\)
\(=\sqrt{13+30\left(\sqrt{2}+1\right)}=\sqrt{43+30\sqrt{2}}\)
\(=\sqrt{25+2\cdot5\cdot3\sqrt{2}+18}=\sqrt{\left(5+3\sqrt{2}\right)^2}=5+3\sqrt{2}\)
b/ \(\left(\sqrt{3}-\sqrt{2}\right)\cdot\sqrt{5+2\sqrt{6}}=\sqrt{3\left(5+2\sqrt{6}\right)}-\sqrt{2\left(5+2\sqrt{6}\right)}\)
\(=\sqrt{15+6\sqrt{6}}-\sqrt{10+4\sqrt{6}}\)
\(=\sqrt{\left(3+\sqrt{6}\right)^2}-\sqrt{\left(2+\sqrt{6}\right)^2}\)
\(=3+\sqrt{6}-2-\sqrt{6}=1\)
c/ \(\sqrt{5-\sqrt{13+4\sqrt{3}}}+\sqrt{3+\sqrt{13+4\sqrt{3}}}\)
\(=\sqrt{5-\sqrt{\left(1+2\sqrt{3}\right)^2}}+\sqrt{3+\sqrt{\left(1+2\sqrt{3}\right)^2}}\)
\(=\sqrt{5-1-2\sqrt{3}}+\sqrt{3+1+2\sqrt{3}}\)
\(=\sqrt{4-2\sqrt{3}}+\sqrt{4+2\sqrt{3}}\)
\(=\sqrt{\left(\sqrt{3}-1\right)^2}+\sqrt{\left(1+\sqrt{3}\right)^2}\)
\(=\sqrt{3}-1+1+\sqrt{3}=2\sqrt{3}\)
d/ \(\sqrt{1+\sqrt{3+\sqrt{13+4\sqrt{3}}}}+\sqrt{1-\sqrt{3-\sqrt{13-4\sqrt{3}}}}\)
\(=\sqrt{1+\sqrt{3+\sqrt{\left(1+2\sqrt{3}\right)^2}}}+\sqrt{1-\sqrt{3-\sqrt{\left(2\sqrt{3}-1\right)^2}}}\)
\(=\sqrt{1+\sqrt{4+2\sqrt{3}}}+\sqrt{1-\sqrt{4-2\sqrt{3}}}\)
\(=\sqrt{1+\sqrt{\left(\sqrt{3}+1\right)^2}}+\sqrt{1-\sqrt{\left(\sqrt{3}-1\right)^2}}\)
\(=\sqrt{1+\sqrt{3}+1}+\sqrt{1-\sqrt{3}+1}\)
\(=\sqrt{2+\sqrt{3}}+\sqrt{2-\sqrt{3}}\)
