Áp dụng bđt AM-GM cho 2 số không âm ta có:
\(\dfrac{1}{\sqrt{1.2006}}>\dfrac{1}{\dfrac{1+2006}{2}}=\dfrac{2}{2007}\)
TT: \(\dfrac{1}{\sqrt{2.2005}}>\dfrac{2}{2007}\)
...
\(\dfrac{1}{\sqrt{2006.1}}>\dfrac{2}{2007}\)
Cộng vế với vế ta được:
\(S>\dfrac{2}{2007}.2006\)
\(\left(\dfrac{1}{3}\sqrt{\dfrac{1}{2}}-\dfrac{2}{3}\sqrt{\dfrac{3}{2}}+\dfrac{2}{7}\sqrt{\dfrac{1}{6}}\right):\left(\dfrac{2}{7}\sqrt{\dfrac{1}{8}}\right)\)
So sánh: \(\dfrac{1}{\sqrt{1.2005}} + \dfrac{1}{\sqrt{2.2004}} +...+ \dfrac{1}{\sqrt{2005.1}} và \dfrac{2005}{1003}\)
Cho bt B=\(\left(\dfrac{1}{\sqrt{x}+1}+\dfrac{1}{\sqrt{x}-1}\right)\left(\dfrac{\sqrt{x}-1}{\sqrt{x}}\right)\)
A)\(\sqrt{2-\sqrt{3}}.\left(\sqrt{6}+\sqrt{2}\right)\)
B)\(\left(\sqrt{2}+1^{ }\right)^3-\left(\sqrt{2}-1\right)^3\) C)\(\dfrac{2\sqrt{8}-\sqrt{12}}{\sqrt{18}-\sqrt{48}}-\dfrac{\sqrt{5}+\sqrt{27}}{\sqrt{30}+\sqrt{162}}\) D)\(\sqrt{\dfrac{2-\sqrt{3}}{2+\sqrt{3}}}+\sqrt{\dfrac{2+\sqrt{3}}{2-\sqrt{3}}}\) E)\(\dfrac{\sqrt{3-\sqrt{5}}.\left(3+\sqrt{5}\right)}{\sqrt{10}+\sqrt{2}}\) F)\(\dfrac{1}{\sqrt{2}+\sqrt{2+\sqrt{3}}}+\dfrac{1}{\sqrt{2}-\sqrt{2-\sqrt{3}}}\)
rút gọn biểu thức
Q=\(\left(\dfrac{\sqrt{a}+1}{\sqrt{a}-2}-\dfrac{\sqrt{a}+2}{\sqrt{a}-1}\right)\):(\(\left(\dfrac{1}{\sqrt{a}-1}-\dfrac{1}{\sqrt{a}}\right)\)
Cho E= \(\left(\dfrac{\sqrt{x}}{\sqrt{x}-1}-\dfrac{1}{x-\sqrt{x}}\right):\left(\dfrac{1}{\sqrt{x}+1}+\dfrac{2}{x-1}\right)\)
a) ĐKXĐ E
b) Rút gọn
Thực hiện phép tính.
a) \(\left(\sqrt{ab}+2\sqrt{\dfrac{b}{a}}-\sqrt{\dfrac{a}{b}+\sqrt{\dfrac{1}{ab}}}\right)\sqrt{ab}\)
b) \(\left(\dfrac{am}{b}\sqrt{\dfrac{n}{m}}-\dfrac{ab}{n}\sqrt{mn}+\dfrac{a^2}{b^2}\sqrt{\dfrac{m}{n}}\right).a^2b^2.\sqrt{\dfrac{n}{m}}\)
Giải chi tiết ra hộ mình với ạ, mình cảm ơn ạ.
\(\left(\sqrt{xy}+2\sqrt{\dfrac{y}{x}}-\sqrt{\dfrac{x}{y}+\sqrt{\dfrac{1}{xy}}}\right):\dfrac{1}{\sqrt{xy}}\)
Cho P =\(\left(\dfrac{3\sqrt{x}+x+2}{\left(\sqrt{x}+2\right).\left(\sqrt{x}+1\right)}-\dfrac{x+\sqrt{x}}{x-1}\right):\left(\dfrac{1}{\sqrt{x}+1}-\dfrac{1}{\sqrt{x}-1}\right)\)
a) Tìm ĐKXĐ, rút gọn
b) Tìm x khi P=-4
c) Tìm P khi x =\(8-2\sqrt{15}\)