Rút gọn các biểu thức
\(A=\left(\frac{\sqrt{a}-2}{\sqrt{a}+2}-\frac{\sqrt{a}+2}{\sqrt{a}-2}\right)\left(\sqrt{a}-\frac{4}{\sqrt{a}}\right)\)
\(B=\frac{1}{1-\sqrt{a}}+\frac{a\sqrt{a}}{\sqrt{a}-1}\)
Cho a,b > 0, a\(\ne\)b
C/m : \(\frac{a+b}{2}>\frac{\left(a-b\right)^2}{4\left(\sqrt{a}-\sqrt{b}\right)}>\sqrt{ab}\)
Bài 1 :Chứng minh các đẳng thức :
a ) \(2\sqrt{2}\left(\sqrt{3}-2\right)\) + \(\left(1+2\sqrt{2}\right)^2-2\sqrt{6}=9\)
b ) \(\sqrt{2+\sqrt{3}}+\sqrt{2-\sqrt{3}}=\sqrt{6}\)
c ) \(\sqrt{11-6\sqrt{2}}+\sqrt{11+6\sqrt{2}}=6\)
Bài 2 : Rút gọn các biểu thức sau :
a ) \(\frac{1}{\sqrt{5}+\sqrt{3}}-\frac{1}{\sqrt{5}-\sqrt{3}}\)
b ) \(\frac{\sqrt{4-2\sqrt{3}}}{\sqrt{6}-\sqrt{2}}\)
c ) \(\frac{1}{2+\sqrt{3}}+\frac{\sqrt{2}}{\sqrt{6}}-\frac{2}{3+\sqrt{3}}\)
Bài 3 : Rút gọn các biểu thức sau :
a ) \(\sqrt{20}-\sqrt{45}+3\sqrt{18}+\sqrt{72}\)
b ) \(\left(\sqrt{28}-2\sqrt{3}+\sqrt{7}\right)\sqrt{7}+\sqrt{84}\)
c ) \(\left(\sqrt{6}+\sqrt{5}\right)^2-\sqrt{120}\)
d ) \(\left(\frac{1}{2}\sqrt{\frac{1}{2}}-\frac{3}{2}\sqrt{2}+\frac{4}{5}\sqrt{200}\right):\frac{1}{8}\)
rút gọn biểu thức
a) A= \(2\sqrt{\frac{1}{2}}+\sqrt{18}\)
b) B= \(\frac{5+3\sqrt{5}}{\sqrt{5}}+\frac{3+\sqrt{3}}{\sqrt{3}+1}-\left(\sqrt{5+3}\right)\)
c) C= \(\frac{1}{x+\sqrt{x}}+\frac{2\sqrt{x}}{x-1}-\frac{1}{x-\sqrt{x}}\left(x>0,x\ne1\right)\)
d) D = \(\left(\frac{\sqrt{x}+2}{x+2\sqrt{x}+1}-\frac{\sqrt{x-2}}{x-1}\right)\left(x+\sqrt{x}\right)\left(x>0,x\ne1\right)\)
e) E = \(\frac{2+\sqrt{3}}{\sqrt{2}+\sqrt{2+\sqrt{3}}}+\frac{2-\sqrt{3}}{\sqrt{2}-\sqrt{2-\sqrt{3}}}\)
a) chứng minh: \(\sqrt{a^2}+\sqrt{b^2}>\sqrt{\left(a+b\right)^2}\)
b) Tìm min của A=\(\sqrt{\left(2021-x\right)^2}+\sqrt{\left(2022-x\right)^2}\)
1.So sánh
a) \(\sqrt{2002}+\sqrt{2004}\) và \(2\sqrt{2003}\)
b)\(\sqrt{4+\sqrt{7}}-\sqrt{4-\sqrt{7}}\) và \(\sqrt{2}\)
2. Rút gọn
a) \(\frac{a^2-\sqrt{a}}{a+\sqrt{a}+1}-\frac{a^2+\sqrt{a}}{a-\sqrt{a}+1}\) với 0 ≤ a ≥ 1
b) \(\frac{a\sqrt{b}-b\sqrt{a}}{\sqrt{a}-\sqrt{b}}-\sqrt{ab}\)
c) \(\frac{\sqrt{a}+\sqrt{b}-1}{a+\sqrt{ab}}+\frac{\sqrt{a}-\sqrt{b}}{2\sqrt{ab}}\left(\frac{\sqrt{b}}{a-\sqrt{ab}}+\frac{\sqrt{b}}{a+\sqrt{ab}}\right)\)
d) \(\frac{a+b+2\sqrt{ab}}{\sqrt{a}+\sqrt{b}}-\frac{a-b}{\sqrt{a}-\sqrt{b}}\)
e)\(\frac{\sqrt{a}-1}{a\sqrt{a}-a+\sqrt{a}}:\frac{1}{a^2+\sqrt{a}}\)
3. Giải phương trình
a)\(\frac{\sqrt{27x}}{\sqrt{3}}=6\)
b)\(\sqrt{x+1}=3-\sqrt{x}\)
c) \(\sqrt{2x+1}=2+\sqrt{x-3}\)
d) \(\sqrt{x-5}-\frac{x-14}{3+\sqrt{x-5}}=3\)
Tính M khi \(a=1+3\sqrt{2}\)và \(b=10+\frac{11\sqrt{8}}{3}\)
\(M=\frac{2a\sqrt{a}\left(\sqrt{a}+\sqrt{2a}-\sqrt{3b}\right)+\sqrt{3b}\left(2\sqrt{a}-\sqrt{3b}\right)-2a\sqrt{a}}{a\sqrt{2}+\sqrt{3ab}}\)
Cho biểu thức\(P=\left(\frac{2a+1}{\sqrt{a^3-1}}-\frac{\sqrt{a}}{a+\sqrt{a}+1}\right).\left(\frac{1+\sqrt{a^3}}{1+\sqrt{a}}-\sqrt{a}\right)\)
a)Rút gọn P
b)Xét dấu \(P.\sqrt{1-a}\)
Cho biểu thức \(P=\left(\frac{1}{\sqrt{1+a}}+\sqrt{1-a}\right):\left(\frac{1}{\sqrt{1-a^2}}+1\right)\)
a)Rút gọn P
b)Tính P khi \(a=\frac{\sqrt{3}}{2+\sqrt{3}}\)
