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\(A=\left(\frac{1}{\sqrt{x}-1}+\frac{1}{x-\sqrt{x}}\right):\frac{\sqrt{x}+1}{x-2\sqrt{x}+1}\)
\(=\left[\frac{\sqrt{x}}{\sqrt{x}\left(\sqrt{x}-1\right)}+\frac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}\right]:\frac{\sqrt{x}+1}{\left(\sqrt{x}-1\right)^2}\)
\(=\frac{\sqrt{x}+1}{\sqrt{x}\left(\sqrt{x}-1\right)}.\frac{\left(\sqrt{x}-1\right)^2}{\sqrt{x}+1}=\frac{\sqrt{x}-1}{\sqrt{x}}=1-\frac{1}{\sqrt{x}}< 1\)
\(a,\)\(T=\frac{x\sqrt{x}-1}{x-\sqrt{x}}-\frac{x\sqrt{x}+1}{x+\sqrt{x}}+\frac{x+1}{\sqrt{x}}\)
\(=\frac{\sqrt{x}^3-1}{\sqrt{x}\left(\sqrt{x}-1\right)}-\)\(\frac{\sqrt{x}^3+1}{\sqrt{x}\left(\sqrt{x}+1\right)}+\frac{x+1}{\sqrt{x}}\)
\(=\frac{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)}\)\(-\frac{\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}+1\right)}+\frac{x+1}{\sqrt{x}}\)
\(=\frac{x+\sqrt{x}+1-x+\sqrt{x}-1+x+1}{\sqrt{x}}\)
\(=\frac{x+2\sqrt{x}+1}{\sqrt{x}}=\frac{\left(\sqrt{x}+1\right)^2}{\sqrt{x}}\)
a)\(M=\left(\frac{\sqrt{x}}{\sqrt{x}-1}-\frac{x}{x-1}\right):\left(\sqrt{x}-\frac{\sqrt{x}}{\sqrt{x}+1}\right)\) \(\left(x>0;x\ne1\right)\)
\(=\left(\frac{\sqrt{x}}{\sqrt{x}-1}-\frac{x}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right):\left(\frac{\sqrt{x}\left(\sqrt{x}+1\right)}{\sqrt{x}+1}-\frac{\sqrt{x}}{\sqrt{x}+1}\right)\)
\(=\left(\frac{\sqrt{x}\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x+1}\right)}-\frac{x}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right):\left(\frac{x+\sqrt{x}}{\sqrt{x}+1}-\frac{\sqrt{x}}{\sqrt{x}+1}\right)\)
\(=\left(\frac{x+\sqrt{x}}{\left(\sqrt{x}-1\right)\left(\sqrt{x+1}\right)}-\frac{x}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right):\frac{x}{\sqrt{x}+1}\)
\(=\frac{\sqrt{x}}{\left(\sqrt{x}-1\right)\left(\sqrt{x+1}\right)}\cdot\frac{\sqrt{x}+1}{x}\)
\(=\frac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}\)
không có giá trị nào của x thỏa mãn M ≤ 0 ...chưa rg đã biết còn mẫu r...mà mẫu thì sao bằng 0 được.... có khi nào sai đề không ....
<=> \((\frac{\sqrt{x}}{\sqrt{x}-1}-\frac{\sqrt{x}}{\sqrt{x}(\sqrt{x}-1)}).\frac{x-1}{\sqrt{x}+1}\)
<=> \(\frac{x-\sqrt{x}}{\sqrt{x}(\sqrt{x}-1)}.\frac{x-1}{\sqrt{x}+1}\)
<=> \(\frac{\sqrt{x}\left(\sqrt{x}-1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)}.\frac{x-1}{\sqrt{x}+1}\)
<=>\(\frac{x-1}{\sqrt{x}+1}\Leftrightarrow\sqrt{x}-1\)
Để Q >0 thì \(\sqrt{x}-1>0\)
\(\sqrt{x}>1\Leftrightarrow x>1\)
Vậy để Q>o thì x>1
Theo đề bài ta có x > 0 nên \(\sqrt{x}>0\)
=> \(\frac{2}{\sqrt{x}}>0\Rightarrow-\frac{2}{\sqrt{x}}< 0\Rightarrow1-\frac{2}{\sqrt{x}}< 1\)
Ta có
M = \(\frac{\sqrt{x}-2}{\sqrt{x}}=\:1-\frac{2}{\sqrt{x}}< 1\)
a) \(\frac{\sqrt{4mn^2}}{\sqrt{20m}}=\sqrt{\frac{4mn^2}{20m}}=\sqrt{\frac{n^2}{5}}=\frac{n}{\sqrt{5}}\)
b) \(\frac{\sqrt{16a^4b^6}}{\sqrt{12a^6b^6}}=\sqrt{\frac{16a^4b^6}{12a^6b^6}}=\sqrt{\frac{4}{3a^2}}=\frac{2}{\sqrt{3}.\left|a\right|}=-\frac{2}{a\sqrt{3}}\)
d) \(\frac{x\sqrt{x}-y\sqrt{y}}{\sqrt{x}-\sqrt{y}}=\frac{\left(\sqrt{x}-\sqrt{y}\right)\left(x+\sqrt{xy}+y\right)}{\sqrt{x}-\sqrt{y}}=x+\sqrt{xy}+y\)
e) \(\sqrt{\frac{x-2\sqrt{x}+1}{x+2\sqrt{x}+1}}=\sqrt{\frac{\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}+1\right)^2}}=\frac{\left|\sqrt{x}-1\right|}{\sqrt{x}+1}\)