Rút gọn biểu thức:
E=\(\left(\frac{\sqrt{x}}{\sqrt{x}-1}-\frac{1}{x-\sqrt{x}}\right):\left(\frac{1}{\sqrt{x+1}}+\frac{2}{x-1}\right)\)
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S
12 tháng 6 2019
\(=\left(\frac{x}{2\sqrt{x}}-\frac{1}{2\sqrt{x}}\right)\left(\frac{x\sqrt{x}}{\sqrt{x}+1}-\frac{x+\sqrt{x}}{\sqrt{x}-1}\right)\)
\(=\left(\frac{x-1}{2\sqrt{x}}\right)\left(\frac{x\sqrt{x}\left(\sqrt{x}-1\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}-\frac{\left(x+\sqrt{x}\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\right)\)
\(=\left(\frac{x-1}{2\sqrt{x}}\right)\left(\frac{x^2-x\sqrt{x}}{x-1}-\frac{x\sqrt{x}+2x+\sqrt{x}}{x-1}\right)\)
\(=\left(\frac{x-1}{2\sqrt{x}}\right)\left(\frac{x^2-2x\sqrt{x}-2x-\sqrt{x}}{x-1}\right)=\frac{x^2-\sqrt{x}-2x\sqrt{x}-2x}{2\sqrt{x}}=\frac{x\sqrt{x}-1-2x-2\sqrt{x}}{2}\)
12 tháng 6 2019
\(\left(\frac{\sqrt{x}}{2}-\frac{1}{2\sqrt{x}}\right)\left(\frac{x\sqrt{x}}{\sqrt{x}+1}-\frac{x+\sqrt{x}}{\sqrt{x}-1}\right)\)
\(=\frac{x-1}{2\sqrt{x}}.\frac{x\sqrt{x}\left(\sqrt{x}-1\right)-\left(x+\sqrt{x}\right)\left(\sqrt{x}+1\right)}{x-1}\)
\(=\frac{x^2-x\sqrt{x}-\left(x\sqrt{x}+x+x+\sqrt{x}\right)}{2\sqrt{x}}\)
\(=\frac{x^2-x\sqrt{x}-x\sqrt{x}-2x-\sqrt{x}}{2\sqrt{x}}\)
\(=\frac{x^2-2x\sqrt{x}-2x-\sqrt{x}}{2\sqrt{x}}\)
29 tháng 6 2019
ĐK : x>0, x khác 1
\(A=\left(\frac{1}{\sqrt{x}+1}+\frac{2\left(1-\sqrt{x}\right)}{x\left(\sqrt{x}+1\right)-\left(\sqrt{x}+1\right)}\right):\left(\frac{\sqrt{x}+2}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}-\frac{2}{x-1}\right)\)
\(=\left(\frac{1}{\sqrt{x}+1}-\frac{2\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)^2}\right):\left(\frac{1}{\sqrt{x}-1}-\frac{2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right)\)
\(=\frac{\sqrt{x}+1-2}{\left(\sqrt{x}+1\right)^2}:\frac{\sqrt{x}+1-2}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}=\frac{\sqrt{x}-1}{\left(\sqrt{x}+1\right)^2}.\frac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}{\sqrt{x}-1}\)
\(=\frac{\sqrt{x}-1}{\sqrt{x}+1}\)
3 tháng 6 2019
ĐK:x>1
M=\(\frac{x-1}{2x}\) .\(\frac{\left(x-\sqrt{x}\right)\left(\sqrt{x}-1\right)-\left(x+\sqrt{x}\right)\left(\sqrt{x}+1\right)}{x-1}\)
=\(\frac{x-1}{2x}\).\(\frac{x\sqrt{x}-x-x+\sqrt{x}-x\sqrt{x}-x-x-\sqrt{x}}{x-1}\)=\(\frac{x-1}{2x}\).\(\frac{-4x}{x-1}\)=-2
Vậy M=-2
25 tháng 8 2020
ĐKXĐ: \(x\ge1\); x khác 2; 3
Ta có:
\(\frac{1}{\sqrt{x}-\sqrt{x-1}}=\frac{\sqrt{x}+\sqrt{x-1}}{x-\left(x-1\right)}=\sqrt{x}+\sqrt{x-1}\)
\(\frac{x-3}{\sqrt{x-1}-\sqrt{2}}=\frac{\left(x-3\right)\left(\sqrt{x-1}+\sqrt{2}\right)}{x-1-2}=\sqrt{x-1}+\sqrt{2}\)
=> \(\frac{1}{\sqrt{x}-\sqrt{x-1}}-\frac{x-3}{\sqrt{x-1}-\sqrt{2}}=\sqrt{x}+\sqrt{x-1}-\left(\sqrt{x-1}+\sqrt{2}\right)=\sqrt{x}-\sqrt{2}\)
\(\frac{2}{\sqrt{2}-\sqrt{x}}-\frac{\sqrt{x}+\sqrt{2}}{\sqrt{2x}-x}=\frac{2\sqrt{x}-\sqrt{x}-2}{\sqrt{x}\left(\sqrt{2}-\sqrt{x}\right)}=\frac{\sqrt{x}-2}{\sqrt{x}\left(\sqrt{2}-\sqrt{x}\right)}\)
=> \(P=\left(\sqrt{x}-\sqrt{2}\right).\frac{\sqrt{x}-2}{\sqrt{x}\left(\sqrt{2}-\sqrt{x}\right)}=\frac{2-\sqrt{x}}{\sqrt{x}}\)
NV
24 tháng 7 2016
\(Q=\left[\frac{\sqrt{x}^3-1}{\sqrt{x}\left(\sqrt{x}-1\right)}-\frac{\sqrt{x}^3+1}{\sqrt{x}\left(\sqrt{x}+1\right)}\right]:\frac{2\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\)
\(=\left[\frac{\left(\sqrt{x}-1\right)\left(x^2+x+1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)}-\frac{\left(\sqrt{x}+1\right)\left(x^2-x+1\right)}{\sqrt{x}\left(\sqrt{x}+1\right)}\right]:\frac{2\left(\sqrt{x}-1\right)}{\sqrt{x}+1}\)
\(=\left(\frac{x^2+x+1}{\sqrt{x}}-\frac{x^2-x+1}{\sqrt{x}}\right).\frac{\sqrt{x}+1}{2\left(\sqrt{x}-1\right)}\)
\(=\frac{2x}{\sqrt{x}}.\frac{\sqrt{x}+1}{2\left(\sqrt{x}-1\right)}=\frac{\sqrt{x}\left(\sqrt{x}+1\right)}{\sqrt{x}-1}\)
sửa đề : \(E=\left(\frac{\sqrt{x}}{\sqrt{x}-1}-\frac{1}{x-\sqrt{x}}\right):\left(\frac{1}{\sqrt{x}+1}+\frac{2}{x-1}\right)\)ĐK : \(x>0;x\ne1\)
\(=\left(\frac{x-1}{\sqrt{x}\left(\sqrt{x}-1\right)}\right):\left(\frac{\sqrt{x}+1}{x-1}\right)\)
\(=\frac{\left(x-1\right)^2}{\sqrt{x}\left(x-1\right)}=\frac{x-1}{\sqrt{x}}\)