a/ \(\sqrt{5}-\sqrt{9-4\sqrt{5}}=\sqrt{5}-\sqrt{\left(\sqrt{5}-2\right)^2}=\sqrt{5}-\sqrt{5}+2=2\)
b/ \(\sqrt{3+\sqrt{5}}-\sqrt{3-\sqrt{5}}=\frac{\sqrt{6+2\sqrt{5}}-\sqrt{6-2\sqrt{5}}}{\sqrt{2}}\)
\(=\frac{\sqrt{\left(\sqrt{5}+1\right)^2}-\sqrt{\left(\sqrt{5}-1\right)^2}}{\sqrt{2}}=\frac{\sqrt{5}+1-\sqrt{5}+1}{\sqrt{2}}=\sqrt{2}\)
Thực hiện các phép tính sau:
\(\sqrt{24+8\sqrt{5}}+\sqrt{9-4\sqrt{5}}\)
\(\sqrt{\sqrt{5}-\sqrt{3-\sqrt{29-12\sqrt{5}}}}\)
\(\sqrt{13+30\sqrt{2+\sqrt{9+4\sqrt{2}}}}\)
\(\sqrt{5-\sqrt{13+4\sqrt{3}}}+\sqrt{3+\sqrt{13+4\sqrt{3}}}\)
\(\sqrt{1+\sqrt{3+\sqrt{13+4\sqrt{3}}}}+\sqrt{1-\sqrt{3-\sqrt{13-4\sqrt{3}}}}\)
Tính:
i) \(\sqrt{8-3\sqrt{7}}+\sqrt{4-\sqrt{7}}\)
j) \(\sqrt{5+\sqrt{21}}-\sqrt{5-\sqrt{21}}\)
k) \(\sqrt{9-3\sqrt{5}}-\sqrt{9+3\sqrt{5}}\)
Thực hiện phép tính (rút gọn biểu thức)
a) \(\sqrt{9+4\sqrt{5}}\) - \(\sqrt{9-4\sqrt{5}}\)
b) \(\sqrt{12-6\sqrt{3}}\) + \(\sqrt{12+6\sqrt{3}}\)
c) \(\sqrt{6\sqrt{2}+11}\) - \(\sqrt{11-6\sqrt{2}}\)
Chứng minh đẳng thức sau:
\(\frac{a+\sqrt{2+\sqrt{5}}.\sqrt{\sqrt{9-4\sqrt{5}}}}{\sqrt[3]{2-\sqrt{5}}.\sqrt[3]{\sqrt{9+4\sqrt{5}}-\sqrt[3]{a^2}}+\sqrt[3]{a}}=-\sqrt[3]{a-1}\)
Chứng minh bất đẳng thức sau:
\(\left(\sqrt[3]{\sqrt{9+4\sqrt{5}}+\sqrt[3]{2+\sqrt{5}}}\right).\sqrt[3]{\sqrt{5-2}}-2,1< 0\)
Rút gọn biểu thức :
a) A=\(\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}\).
b)B=\(\sqrt[3]{20+14\sqrt{2}}+\sqrt[3]{20-14\sqrt{2}}\)
c) C=\(\sqrt[3]{9+4\sqrt{5}}+\sqrt[3]{9-4\sqrt{5}}.\)
Rút gọn: A = \(\frac{a+\sqrt{2+\sqrt{5}}.\sqrt{\sqrt{9-4\sqrt{5}}}}{\sqrt[3]{2-\sqrt{5}}.\sqrt[3]{\sqrt{9-4\sqrt{5}}}-\sqrt[3]{a^2}+\sqrt[3]{a}}\)
tính ;\(\sqrt{2-\sqrt[3]{3+\sqrt[4]{4-\sqrt[5]{5+\sqrt[6]{6-\sqrt[7]{7+\sqrt[8]{8-\sqrt[9]{9}}}}}}}}\)
Tính
\(\frac{\sqrt{10}+5\sqrt{3}}{\sqrt{15}+\sqrt{5}}-\frac{3}{2\sqrt{2}-\sqrt{5}}+\sqrt{9+4\sqrt{2}}\)
Tính
\(\dfrac{4}{\sqrt{5}-3}\)-\(\dfrac{4}{\sqrt{5}+3}\)
Giải phương trình
\(\sqrt{4x-20}\)-3\(\sqrt{\dfrac{x-5}{9}}\)=\(\sqrt{1-x}\)
\(Tính\)
\(a.\left(\sqrt{10}-\sqrt{2}\right)\left(3+\sqrt{5}\right)\sqrt{27-9\sqrt{5}}\)
\(b.\sqrt{\frac{3\sqrt{3}-4}{2\sqrt{3}+1}}-\sqrt{\frac{4+\sqrt{3}}{5-2\sqrt{3}}}\)
\(c.\frac{3-4\sqrt{3}}{\sqrt{6}-\sqrt{2}-\sqrt{5}}\)
\(d.\left(\sqrt{11}-\sqrt{3}\right)\left(\sqrt{13-\sqrt{6}+2\sqrt{30-\sqrt{45}}}+\sqrt{11}-\sqrt{10-\sqrt{6}}\right)\)
\(e.\frac{\sqrt{4+\sqrt{5}}+\sqrt{4-\sqrt{5}}}{\sqrt{4}+\sqrt{11}}-\frac{\sqrt{20-4\sqrt{23}}}{\sqrt{5+\sqrt{2}}-\sqrt{5-\sqrt{2}}}\)
