\(\left(\frac{\sqrt{a}}{\sqrt{a}-1}+\frac{\sqrt{a}}{a-1}\right):\left(\frac{2}{a}-\frac{2-a}{a\sqrt{a}+a}\right)\)
=\(\frac{\sqrt{a}\left(\sqrt{a}+1\right)+\sqrt{a}}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}:\frac{2\left(\sqrt{a}+1\right)-2+a}{a\left(\sqrt{a}+1\right)}\)
= \(\frac{\sqrt{a}\left(\sqrt{a}+2\right)}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}.\frac{a\left(\sqrt{a+1}\right)}{\sqrt{a}\left(2+\sqrt{a}\right)}\)
=\(\frac{a}{\sqrt{a}-1}\)
Rút gọn:
a) \(\frac{a-b}{\sqrt{a}-\sqrt{b}}\)-\(\frac{\sqrt{a^3}-\sqrt{b^3}}{a-b}\)(\(a\ge0\),\(b\ge0\),\(a\ne b\))
b)\(\frac{\left(\sqrt{a}-\sqrt{b}\right)^2-4\sqrt{ab}}{\sqrt{a}-\sqrt{b}}-\frac{a\sqrt{b}+b\sqrt{a}}{\sqrt{ab}}\)\(\left(a>0,b>0,a\ne b\right)\)
C)\(\left(\frac{1}{\sqrt{a}-1}-\frac{1}{\sqrt{a}}\right)\div\left(\frac{\sqrt{a}+1}{\sqrt{a}-2}-\frac{\sqrt{a}+2}{\sqrt{a}-1}\right)\)\(\left(a>0,a\ne1,a\ne4\right)\)
d)\(\left(\frac{a\sqrt{a}+b\sqrt{b}}{\sqrt{a}+\sqrt{b}}-\sqrt{ab}\right)\)\(\left(\frac{\sqrt{a}+\sqrt{b}}{a-b}\right)^2\)\(\left(a>0,b>0,a\ne b\right)\)
e)\(\left(\frac{\sqrt{x}}{3+\sqrt{x}}+\frac{x+9}{9-x}\right)\):\(\left(\frac{3\sqrt{x}+1}{x-3\sqrt{x}}-\frac{1}{\sqrt{x}}\right)\)\(\left(x>0,x\ne9\right)\)
1 Rút gọn:
a) A=\(\frac{\sqrt[]{2+\sqrt[]{3}}}{4}+\sqrt[]{\frac{2-\sqrt[]{3}}{16}}+\frac{1}{\sqrt[]{3}+\sqrt[]{2}+1}\)
b)\(\left(\sqrt[]{a+\sqrt[]{a^2-8}}\right).\left(\sqrt[]{a-2\sqrt[]{2}}-\sqrt[]{a+2\sqrt[]{2}}\right),a>=2\sqrt[]{2}\)
2.Cho x= \(\sqrt[]{2-\sqrt[]{3}}.\left(\sqrt[]{6}+\sqrt[]{2}\right)-\frac{2\sqrt[]{6}+\sqrt[]{3}}{\sqrt[]{8}+1}\). Tính A= \(x^5-3x^4-3x^3+6x^2-20x+2022\)
3. Cho a,b,c >0, \(\frac{a}{a+b}=\frac{b}{c+a}=\frac{c}{a+b}\). CMR: \(\frac{\left(a+b\right)^3}{c^3}+\frac{\left(b+c\right)^3}{a^3}+\frac{\left(a+c\right)^3}{b^3}+24\)
A=\(\left(\frac{1}{\sqrt{a}-1}-\frac{2\sqrt{a}}{a\sqrt{a}+\sqrt{a}-a-1}\right):\frac{\sqrt{a}-2}{a+1}\left(a\ge0;a\ne1\right)\)
Rút gọn
\(P=\left(\frac{1}{a-1}+\frac{3\sqrt{a}+5}{a\sqrt{a}-a-\sqrt{a}+1}\right).\frac{\left(\sqrt{a}+1\right)^2}{4\sqrt{a}}\left(a>0,a\ne1\right)\)
a)Rút gọn
b)Đặt Q= \(\left(a-\sqrt{a}+1\right)P\).Chứng minh Q>1
A= \(\left(\frac{1}{\sqrt{x}+1}-\frac{2\sqrt{x}-2}{x\sqrt{x}-\sqrt{x}+x-1}\right):\left(\frac{1}{\sqrt{x}-1}-\frac{2}{x-1}\right)\)
a) Rút gọn A
b) Tìm GTNN của A
Cho: A=\(\left(\frac{\sqrt{x}+1}{\sqrt{x}-1}+\frac{1-\sqrt{x}}{\sqrt{x}+1}\right):\left(\frac{\sqrt{x}+1}{\sqrt{x}-1}+\frac{\sqrt{x}}{\sqrt{x}+1}+\frac{\sqrt{x}}{1-x}\right)\)
So sánh A với 2
1. Rút gọn biểu thức: A= \(\left(\sqrt{7-4\sqrt{3}}-\frac{\sqrt{15}-3}{\sqrt{3}}\right).\left(2+\sqrt{5}\right)\)
2. Cho biểu thức: M= \(\left(\frac{x+2}{x\sqrt{x}-1}+\frac{\sqrt{x}}{x+\sqrt{x}+1}-\frac{1}{\sqrt{x}-1}\right):\frac{\sqrt{x}-1}{2}\)( với x \(\ge\)0, x\(\ne\)1)
a, Rút gọn biểu thức M
b, Tìm x để M=2
3.
a, Rút gọn biểu thức: \(\frac{4}{\sqrt{5}-\sqrt{3}}-\sqrt{20}-\sqrt{27}\)
b, Với a > 1, cho biểu thức P= \(\left(\frac{2}{\sqrt{a+1}}+\sqrt{a-1}\right):\left(\frac{2}{\sqrt{a^2-1}}+1\right)\)
Rút gọn biểu thức P, tìm giá trị của a để P = 2
Cho biểu thức:
\(E=\left(\frac{\sqrt{a}-2}{a-1}-\frac{\sqrt{a}+2}{a+2\sqrt{a}+1}\right):\frac{2}{\left(1-a\right)^2}
\)
a,Rút gọn E
b,Tìm Max E
Cho biểu thức
A = \(\left(1+\frac{\sqrt{a}}{a+1}\right):\left(\frac{1}{\sqrt{a}-1}-\frac{2\sqrt{a}}{a\sqrt{a}+\sqrt{a}-a-1}\right)\)
a. Tìm ĐKXĐ và rút gọn biểu thức
b. Tìm a để A > 1
