\(\sqrt[3]{7+5\sqrt{2}}+\frac{1}{\sqrt[3]{7+5\sqrt{2}}}\\ =\sqrt[3]{\left(\sqrt{2}+1\right)^3}+\frac{1}{\sqrt[3]{\left(\sqrt{2}+1\right)^3}}=\sqrt{2}+1+\frac{1}{\sqrt{2}+1}=2\sqrt{2}\)
a,\(\frac{2\sqrt{2}}{\sqrt{2\sqrt{2}+1}+1}-\frac{2\sqrt{2}}{\sqrt{2\sqrt{2}+1}-1}\)
b,\(\frac{\sqrt{15}-\sqrt{12}}{\sqrt{5}-2}+\frac{1}{2-\sqrt{3}}\)
c,\(\frac{2}{\sqrt{3}-\sqrt{5}}+\frac{3-2\sqrt{3}}{\sqrt{3}-2}\)
d,\(\frac{-4}{\sqrt{7}-\sqrt{5}}+\frac{1}{\sqrt{3}-1}+\frac{4-2\sqrt{5}}{\sqrt{5}-2}\)
e,\(\frac{6}{\sqrt{5}-1}+\frac{7}{1-\sqrt{3}}-\frac{2}{\sqrt{3}-\sqrt{5}}\)
Bài 1: Tính
1, \(A=\left(1-\frac{5+\sqrt{5}}{1+\sqrt{5}}\right).\left(\frac{5-\sqrt{5}}{1-\sqrt{5}}-1\right)\)
2, \(B=\left(\frac{3\sqrt{125}}{15}-\frac{10-4\sqrt{6}}{\sqrt{5}-2}\right).\frac{1}{\sqrt{5}}\)
3, \(C=\left(\frac{\sqrt{1000}}{100}-\frac{5\sqrt{2}-2\sqrt{5}}{2\sqrt{5}-8}\right).\frac{\sqrt{10}}{10}\)
4, \(D=\frac{1}{\sqrt{49+20\sqrt{6}}}-\frac{1}{\sqrt{49-20\sqrt{6}}}+\frac{1}{\sqrt{7-4\sqrt{3}}}\)
5, \(E=\frac{1}{\sqrt{4-2\sqrt{3}}}-\frac{1}{\sqrt{7-\sqrt{48}}}+\frac{3}{\sqrt{14-6\sqrt{5}}}\)
6, \(F=\frac{1}{\sqrt{2}-\sqrt{3}}\sqrt{\frac{3\sqrt{2}-2\sqrt{3}}{3\sqrt{2}+2\sqrt{3}}}\)
7, \(G=\frac{\sqrt{15-10\sqrt{2}}+\sqrt{13+4\sqrt{10}-\sqrt{11-2\sqrt{10}}}}{2\sqrt{3+2\sqrt{2}}+\sqrt{9-4\sqrt{2}+\sqrt{12+8\sqrt{2}}}}\)
Giải :
1) \(\frac{5\sqrt{7}-7\sqrt{5}+2\sqrt{70}}{\sqrt{35}}\)
2) \(\sqrt{\frac{4}{3}}+\sqrt{12}-\frac{4}{3}\sqrt{\frac{3}{4}}\)
3) \(\frac{1}{1+\sqrt{2}+\sqrt{3}}\)
4) \(\left(5\sqrt{\frac{1}{5}}+\frac{1}{2}\sqrt{20}-\frac{5}{4}\sqrt{\frac{4}{5}+\sqrt{5}}\right):2\sqrt{5}\)
rút gọn biểu thức
a) \(\frac{3}{2+\sqrt{3}}+\frac{13}{4-\sqrt{3}}+\frac{6}{\sqrt{3}}\)
b) \(\left(\frac{\sqrt{14}-\sqrt{7}}{\sqrt{2}-1}+\frac{\sqrt{15}-\sqrt{5}}{\sqrt{3}-1}\right):\frac{1}{\sqrt{7}-\sqrt{5}}\)
c) \(\sqrt{\left(2+\sqrt{3}\right)^2}-\sqrt{28-10\sqrt{3}}\)
d) \(\frac{3}{3+2\sqrt{3}}+\frac{3}{3-2\sqrt{3}}\)
e) \(\sqrt{20}-15\sqrt{\frac{1}{5}}+\sqrt{\left(1-\sqrt{5}\right)^2}\)
Tinh
\(a,\sqrt{75}-\sqrt{5\frac{1}{3}}+\frac{9}{2}\sqrt{2\frac{2}{3}}+2\sqrt{27}\)
\(b,\sqrt{48}+\sqrt{5\frac{1}{3}}+2\sqrt{75}-5\sqrt{1\frac{1}{3}}\)
\(c,\left(\sqrt{15}+2\sqrt{3}\right)^2+12\sqrt{5}\)
\(d,\left(\sqrt{6}+2\right)\left(\sqrt{3}-\sqrt{2}\right)\)
\(e,\left(\sqrt{3}+1\right)^2-2\sqrt{3}+4\)
\(f,\frac{1}{7+4\sqrt{3}}+\frac{1}{7-4\sqrt{3}}\)
\(g,\left(\frac{1}{\sqrt{5}-\sqrt{2}}-\frac{1}{\sqrt{5}+\sqrt{2}}+1\right)\frac{1}{\left(\sqrt{2}+1\right)^2}\)
Tính
\(A=\sqrt{20}-3\sqrt{8}+5\sqrt{45}\)
\(B=\dfrac{30}{\sqrt{7}-1}+\dfrac{15}{\sqrt{7}+2}\)
\(C=\left(3-\dfrac{5-\sqrt{5}}{\sqrt{5}-1}\right)\left(3+\dfrac{5+\sqrt{5}}{\sqrt{5}+1}\right)\)
\(D=\sqrt{\left(3-\sqrt{2}\right)^2}-\sqrt{\left(1-\sqrt{2}\right)^2}\)
\(E=\sqrt{7-4\sqrt{3}}-\sqrt{3+2\sqrt{3}}\)
So sánh:
a, \(\frac{3\sqrt{7}+5\sqrt{2}}{\sqrt{5}}\) và \(\sqrt{35}+\sqrt{10}\)
b, \(\frac{\sqrt{3+\sqrt{5}}}{\sqrt{2}}\) và \(\frac{1+\sqrt{5}}{2}\)
c, \(\frac{2+\sqrt{2}}{2-\sqrt{2}}+\frac{2-\sqrt{2}}{2+\sqrt{2}}\) và \(4\sqrt{2}\)
d, \(\sqrt{4+\sqrt{7}}-\sqrt{4-\sqrt{7}}\) và \(\sqrt{3}\)
Rút gọn biểu thức:
1) \(\sqrt{\left(1-\sqrt{2}\right)^2}+\sqrt{\left(\sqrt{2}+3\right)^2}\)
2) \(\sqrt{\left(\sqrt{3}-2\right)^2}+\sqrt{\left(\sqrt{3}-1\right)^2}\)
3) \(\left(\sqrt{19}-3\right)\left(\sqrt{19}+3\right)\)
4) \(4x+\sqrt{\left(x-12\right)^2}\left(x\ge2\right)\)
5) \(\frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}-\sqrt{5}}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}\)
6) \(x+2y-\sqrt{\left(x^2-4xy+4y^2\right)^2}\left(x\ge2y\right)\)
thực hiện phép tính:
1, \(\sqrt{5+2\sqrt{24}}-\sqrt{2}\)
2, \(\frac{3-2\sqrt{3}}{\sqrt{3}-2}\) 3, \(\frac{\sqrt{15}-\sqrt{5}}{1-\sqrt{3}}\)
4, \(\frac{1}{1-\sqrt{2}}-\frac{1}{1+\sqrt{2}}\) 5, \(\frac{\sqrt{12}-6}{\sqrt{8}-\sqrt{24}}-\frac{3-\sqrt{3}}{\sqrt{3}}-\frac{4}{1-\sqrt{7}}\)
