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\(A=\left(\dfrac{1}{\sqrt{x}-1}-\dfrac{1}{\sqrt{x}}\right).\left(\dfrac{\sqrt{x}+1}{\sqrt{x}-2}-\dfrac{\sqrt{x}+2}{\sqrt{x}-1}\right)\text{x > 0, x ≠ 1}\)
\(A=\dfrac{\sqrt{x}-\sqrt{x}+1}{\sqrt{x}\left(\sqrt{x}-1\right)}.\dfrac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)-\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\)
\(A=\dfrac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}.\dfrac{x-1-x+2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\) \(=\dfrac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}.\dfrac{1}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\)
\(A=\dfrac{1}{\sqrt{x}\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)^2}\)
\(=>A=\left(\dfrac{x-1}{\sqrt{x}\left(\sqrt{x}-1\right)}\right).\left[\dfrac{\sqrt{x}+1-2}{x-1}\right]\)
\(=>A=\dfrac{\sqrt{x}+1}{\sqrt{x}}.\dfrac{1}{\sqrt{x}+1}=\dfrac{1}{\sqrt{x}}\)
b,\(=>\dfrac{1}{\sqrt{x}}=\dfrac{1}{2}=>\sqrt{x}=2=>x=\sqrt{2}\left(tm\right)\)
\(A=2\left|2-\sqrt{5}\right|-\dfrac{8\left(3+\sqrt{5}\right)}{\left(3-\sqrt{5}\right)\left(3+\sqrt{5}\right)}\)
\(=2\left(\sqrt{5}-2\right)-\dfrac{8\left(3+\sqrt{5}\right)}{4}=2\sqrt{5}-4-2\left(3+\sqrt{5}\right)\)
\(=2\sqrt{5}-4-6-2\sqrt{5}=-10\)
\(B=\left(\dfrac{2\sqrt{x}}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}-\dfrac{\sqrt{x}-2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\right):\left(\dfrac{\sqrt{x}-2+2}{\sqrt{x}-2}\right)\)
\(=\left(\dfrac{\sqrt{x}+2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\right):\left(\dfrac{\sqrt{x}}{\sqrt{x}-2}\right)\)
\(=\dfrac{1}{\sqrt{x}-2}.\dfrac{\sqrt{x}-2}{\sqrt{x}}=\dfrac{1}{\sqrt{x}}\)
`A=((x\sqrtx-1)/( x-\sqrtx)-(x\sqrtx+1)/(x+\sqrtx)(\sqrtx/(\sqrtx-1)-1/(\sqrtx+1))(x>0,x ne 1)`
`=(((\sqrtx-1)(x+\sqrtx+1))/( x-\sqrtx)-((\sqrtx+1)(x-\sqrtx+1))/(x+\sqrtx)((x+\sqrtx-\sqrtx+1)/(x-1))`
`=((x+\sqrtx+1+x-\sqrtx+1)/\sqrtx) .((x+1) /( x-1)) `
`=((2x+2)/\sqrtx).((x+1) /(x-1 ) )`
`=( 2(x+1)^2) /(\sqrtx(x-1))`
`A=((x\sqrtx-1)/( x-\sqrtx)-(x\sqrtx+1)/(x+\sqrtx)(\sqrtx/(\sqrtx-1)-1/(\sqrtx+1))(x>0,x ne 1)`
`=(((\sqrtx-1)(x+\sqrtx+1))/( x-\sqrtx)-((\sqrtx+1)(x-\sqrtx+1))/(x+\sqrtx))((x+\sqrtx-\sqrtx+1)/(x-1))`
`=((x+\sqrtx+1+x-\sqrtx+1)/\sqrtx) .((x+1) /( x-1)) `
`=((2x+2)/\sqrtx).((x+1) /(x-1 ) )`
`=( 2(x+1)^2) /(\sqrtx(x-1))`
a) \(P=\dfrac{1}{\sqrt{5}-2}+\dfrac{1}{\sqrt{5}+2}=\dfrac{\sqrt{5}+2+\sqrt{5}-2}{\left(\sqrt{5}-2\right)\left(\sqrt{5}+2\right)}=\dfrac{2\sqrt{5}}{\left(\sqrt{5}\right)^2-2^2}=2\sqrt{5}\)
b)\(Q=\left(1+\dfrac{\sqrt{x}+1}{\sqrt{x}-1}\right)\cdot\dfrac{1}{\sqrt{x}}=\dfrac{\sqrt{x}-1+\sqrt{x}+1}{\sqrt{x}-1}\cdot\dfrac{1}{\sqrt{x}}\)
\(Q=\dfrac{2\sqrt{x}}{\sqrt{x}-1}\cdot\dfrac{1}{\sqrt{x}}=\dfrac{2}{\sqrt{x}-1}\)
Tick hộ nha
a.
\(A=\left(1-\dfrac{1}{\sqrt{a}}\right)\left(\dfrac{1}{\sqrt{a}-1}+\dfrac{1}{\sqrt{a}+1}\right)\)
\(=\left(\dfrac{1-\sqrt{a}}{\sqrt{a}}\right)\left(\dfrac{\sqrt{a}-1+\sqrt{a}+1}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}\right)\)
\(=\dfrac{1-\sqrt{a}}{\sqrt{a}}.\dfrac{2\sqrt{a}}{a-1}=\dfrac{2\left(1-\sqrt{a}\right)}{a-1}=\dfrac{-2\left(\sqrt{a}-2\right)}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}\)
\(=\dfrac{-2}{\sqrt{a}+1}\)
b.
\(a-2\sqrt{2}\rightarrow\sqrt{a}=\sqrt{2}-1\)
\(=2-2\sqrt{2}+1\)
=\(\left(\sqrt{2}-1\right)^2\)
\(\rightarrow A=\dfrac{-2}{\sqrt{2}-1+1}=\dfrac{-1}{\sqrt{2}}=\sqrt{2}\)
=>\(A=\left(\dfrac{\sqrt{a}-1}{\sqrt{a}}\right).\left(\dfrac{\sqrt{a}+1+\sqrt{a}-1}{a-1}\right)\left(a>0,a\ne1\right)\)
\(=\dfrac{\sqrt{a}-1}{\sqrt{a}}.\dfrac{2\sqrt{a}}{a-1}=\dfrac{2}{\sqrt{a}+1}\)
b, \(a=3-2\sqrt{2}=\left(\sqrt{2}-1\right)^2\) thế vào A
\(=>A=\dfrac{2}{\sqrt{\left(\sqrt{2}-1\right) ^2}+1}=\dfrac{2}{\sqrt{2}}\)
1.
\(Q=\left[\frac{\sqrt{x}+2}{(\sqrt{x}+1)^2}-\frac{\sqrt{x}-2}{(\sqrt{x}-1)(\sqrt{x}+1)}\right].\sqrt{x}(\sqrt{x}+1)\)
\(=\frac{\sqrt{x}(\sqrt{x}+2)}{\sqrt{x}+1}-\frac{\sqrt{x}(\sqrt{x}-2)}{\sqrt{x}-1}\)
\(=\frac{\sqrt{x}(\sqrt{x}+2)(\sqrt{x}-1)-\sqrt{x}(\sqrt{x}-2)(\sqrt{x}+1)}{(\sqrt{x}-1)(\sqrt{x}+1)}=\frac{2x}{x-1}\)
2.
\(A=\left[\frac{\sqrt{x}+2-(\sqrt{x}-2)}{(\sqrt{x}-2)(\sqrt{x}+2)}-\frac{4\sqrt{x}}{x-4}\right].\frac{x-4}{\sqrt{x}+1}\)
\(=\left(\frac{4}{x-4}-\frac{4\sqrt{x}}{x-1}\right).\frac{x-4}{\sqrt{x}+1}=\frac{4(1-\sqrt{x})}{x-4}.\frac{x-4}{\sqrt{x}+1}=\frac{4(1-\sqrt{x})}{\sqrt{x}+1}\)
Lời giải:
ĐK: $x\geq 0; x\neq 1$
\(A=\left[\frac{(\sqrt{x}-1)(x+2\sqrt{x}+2)}{(\sqrt{x}-1)(\sqrt{x}+1)}-\frac{\sqrt{x}+2}{(\sqrt{x}+1)(\sqrt{x}+2)}\right].\frac{\sqrt{x}-1}{(\sqrt{x}-1)(2\sqrt{x}+3)}\)
\(=\left(\frac{x+2\sqrt{x}+2}{\sqrt{x}+1}-\frac{1}{\sqrt{x}+1}\right).\frac{1}{2\sqrt{x}+3}=\frac{x+2\sqrt{x}+1}{\sqrt{x}+1}.\frac{1}{2\sqrt{x}+3}=\frac{(\sqrt{x}+1)^2}{(\sqrt{x}+1)(2\sqrt{x}+3)}=\frac{\sqrt{x}+1}{2\sqrt{x}+3}\)
Ta có: \(A=\left(\dfrac{2}{\sqrt{x}-1}-\dfrac{\sqrt{x}+1}{x-\sqrt{x}}\right)\left(\dfrac{x+\sqrt{x}}{\sqrt{x}+1}-\dfrac{2\sqrt{x}-2}{\sqrt{x}-1}\right)\)
\(=\left(\dfrac{2\sqrt{x}-\sqrt{x}-1}{\sqrt{x}\left(\sqrt{x}-1\right)}\right)\cdot\left(\sqrt{x}-2\right)\)
\(=\dfrac{\sqrt{x}-1}{\sqrt{x}\left(\sqrt{x}-1\right)}\cdot\left(\sqrt{x}-2\right)\)
\(=\dfrac{\sqrt{x}-2}{\sqrt{x}}\)
\(A=\left(\dfrac{1}{\sqrt{x}-1}-\dfrac{1}{\sqrt{x}}\right).\left(\dfrac{\sqrt{x}+1}{\sqrt{x}-2}-\dfrac{\sqrt{x}+2}{\sqrt{x}-1}\right)\)
\(=\dfrac{\sqrt{x}-\sqrt{x}+1}{\sqrt{x}\left(\sqrt{x}-1\right)}.\dfrac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)-\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}-2\right)}\)
\(=\dfrac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}.\dfrac{3}{\left(\sqrt{x}-1\right)\left(\sqrt{x}-2\right)}=\dfrac{3}{\sqrt{x}\left(\sqrt{x}-1\right)^2\left(\sqrt{x}-2\right)}\)