\(\frac{1}{\sqrt{k\left(k+1\right)}}.\frac{\sqrt{k+1}-\sqrt{k}}{k+1}\) giúp mình rút gọn với ạ
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KG
0
14 tháng 6 2020
Ta có:
\(\frac{1}{\sqrt{k}}=\frac{2}{2\sqrt{k}}=\frac{2}{\sqrt{k}+\sqrt{k}}< \frac{2}{\sqrt{k}+\sqrt{k-1}}=\frac{2\left(\sqrt{k}-\sqrt{k-1}\right)}{\left(\sqrt{k}-\sqrt{k-1}\right)\sqrt{k}+\sqrt{k-1}}\)
\(=\frac{2\left(\sqrt{k}-\sqrt{k-1}\right)}{k-\left(k-1\right)}=2\left(\sqrt{k}-\sqrt{k-1}\right)\)
NV
16 tháng 6 2016
ĐKXĐ: a > 0
a/ \(K=\left[\frac{\sqrt{a}}{\sqrt{a}-1}-\frac{1}{\sqrt{a}\left(\sqrt{a}-1\right)}\right]:\left[\frac{1}{\sqrt{a}-1}+\frac{2}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}\right]\)
\(=\left[\frac{a-1}{\sqrt{a}\left(\sqrt{a}-1\right)}\right]:\left[\frac{\sqrt{a}+3}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}\right]\)
\(=\left[\frac{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}{\sqrt{a}\left(\sqrt{a}-1\right)}\right].\left[\frac{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}{\sqrt{a}+3}\right]\) \(=\frac{\left(\sqrt{a}+1\right)^2\left(\sqrt{a}-1\right)}{\sqrt{a}\left(\sqrt{a}+3\right)}\)
b/ Ta có: \(\sqrt{a}=\sqrt{3+2\sqrt{2}}=\sqrt{\left(\sqrt{2}+1\right)^2}=\sqrt{2}+1\)
\(K=\frac{\left(\sqrt{a}+1\right)^2\left(\sqrt{a}-1\right)}{\sqrt{a}\left(\sqrt{a}+3\right)}=\frac{\left(\sqrt{2}+2\right)\sqrt{2}}{\left(\sqrt{2}+1\right)\left(\sqrt{2}+4\right)}=\frac{2\left(\sqrt{2}+1\right)}{\sqrt{2}\left(\sqrt{2}+1\right)\left(2\sqrt{2}+1\right)}\)
\(=\frac{\sqrt{2}}{2\sqrt{2}+1}\)
c/ \(K< 0\Leftrightarrow\frac{\left(\sqrt{a}+1\right)^2\left(\sqrt{a}-1\right)}{\sqrt{a}\left(\sqrt{a}+3\right)}< 0\)\(\Rightarrow\left(\sqrt{a}+1\right)^2\left(\sqrt{a}-1\right)< 0\)
\(\Rightarrow\sqrt{a}-1< 0\) (vì \(\left(\sqrt{a}+1\right)^2>0\)) \(\Rightarrow\sqrt{a}< 1\Rightarrow a< 1\)
Vậy \(0< a< 1\) thì K < 0
24 tháng 7 2017
a. ĐKXĐ \(x\ge0\)và \(x\ne9\)
Ta có \(K=\left(\frac{2\sqrt{x}}{\sqrt{x}+3}+\frac{\sqrt{x}}{\sqrt{x}-3}-\frac{3\left(\sqrt{x}+3\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}\right):\left(\frac{2\sqrt{x}-2}{\sqrt{x}-3}-1\right)\)
\(=\frac{2\sqrt{x}\left(\sqrt{x}-3\right)+\sqrt{x}\left(\sqrt{x}+3\right)-3\sqrt{x}-9}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}:\frac{2\sqrt{x}-2-\sqrt{x}+3}{\sqrt{x}-3}\)
\(=\frac{2x-6\sqrt{x}+x+3\sqrt{x}-3\sqrt{x}-9}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}:\frac{\sqrt{x}+1}{\sqrt{x}-3}\)
\(=\frac{3x-6\sqrt{x}-9}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}.\frac{\sqrt{x}-3}{\sqrt{x}+1}=\frac{3\left(x-2\sqrt{x}-3\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}.\frac{\sqrt{x}-3}{\sqrt{x}+1}\)
\(=\frac{3\left(\sqrt{x}+1\right)\left(\sqrt{x}-3\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}.\frac{\sqrt{x}-3}{\sqrt{x}+1}=\frac{3\left(\sqrt{x}-3\right)}{\sqrt{x}+3}\)
b. Để \(K< -1\Rightarrow\frac{3\sqrt{x}-9+\sqrt{x}+3}{\sqrt{x}+3}< 0\Rightarrow\frac{4\sqrt{x}-6}{\sqrt{x}+3}< 0\Rightarrow4\sqrt{x}-6< 0\)vì \(\sqrt{x}+3\ge3\)
\(\Rightarrow0\le x< \frac{9}{4}\left(tm\right)\)
Vậy với \(0\le x< \frac{9}{4}\)thì K<-1
c. \(K=\frac{3\sqrt{x}-9}{\sqrt{x}+3}=3+\frac{-18}{\sqrt{x}+3}\)
Ta có \(\sqrt{x}+3\ge3\Rightarrow\frac{1}{\sqrt{x}+3}\le\frac{1}{3}\Rightarrow-\frac{18}{\sqrt{x}+3}\ge-6\Rightarrow3+\frac{-18}{\sqrt{x}+3}\ge-3\)
\(\Rightarrow K\ge-3\)
Vậy \(MinK=-3\Leftrightarrow\sqrt{x}=0\Leftrightarrow x=0\)
7 tháng 10 2018
Ta thấy: k thuộc N* nên \(\sqrt{k+1}>\sqrt{k}\)
\(\Rightarrow\frac{1}{\left(k+1\right)\sqrt{k}}=\frac{2}{\left(2\sqrt{k+1}\right).\left(\sqrt{k+1}.\sqrt{k}\right)}< \frac{2}{\left(\sqrt{k+1}.\sqrt{k}\right).\left(\sqrt{k+1}+\sqrt{k}\right)}\)
\(=\frac{2\left(\sqrt{k+1}-\sqrt{k}\right)}{\left(\sqrt{k+1}.\sqrt{k}\right)\left(k+1-k\right)}=2\left(\frac{1}{\sqrt{k}}-\frac{1}{\sqrt{k+1}}\right)\)
\(\Rightarrow\frac{1}{\left(k+1\right)\sqrt{k}}< 2\left(\frac{1}{\sqrt{k}}-\frac{1}{\sqrt{k+1}}\right)\)(đpcm).
3 tháng 1 2016
\(\Rightarrow C=1+\left[\frac{\left(2\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\left(1-\sqrt{x}\right)\left(\sqrt{x}+1\right)}-\frac{2x\sqrt{x}-\sqrt{x}+x}{\left(1-\sqrt{x}\right)\left(1+\sqrt{x}+x\right)}\right].\frac{x-\sqrt{x}}{2\sqrt{x}-1}\)
\(=1+\left[\frac{\left(2\sqrt{x}-1\right)\left(1+\sqrt{x}+x\right)-\left(2x\sqrt{x}-\sqrt{x}+x\right)}{\left(1-\sqrt{x}\right)\left(1+\sqrt{x}+x\right)}\right].\frac{x-\sqrt{x}}{2\sqrt{x}-1}\)
\(=1+\left[\frac{2\sqrt{x}+2x+2x\sqrt{x}-1-\sqrt{x}-x-2x\sqrt{x}+\sqrt{x}-x}{\left(1-\sqrt{x}\right)\left(1+\sqrt{x}+x\right)}\right].\frac{\sqrt{x}\left(\sqrt{x}-1\right)}{2\sqrt{x}-1}\)
\(=1+\left[\frac{2\sqrt{x}-1}{\left(1-\sqrt{x}\right)\left(1+\sqrt{x}+x\right)}\right].-\frac{\sqrt{x}\left(1-\sqrt{x}\right)}{2\sqrt{x}-1}\)
\(=1-\frac{\sqrt{x}}{1+\sqrt{x}+x}\) \(=\frac{1+\sqrt{x}+x-\sqrt{x}}{1+\sqrt{x}+x}=\frac{1+x}{1+\sqrt{x}+x}\)
1 tháng 7 2015
đk: a>0; a khác 1
\(K=\left(\frac{\sqrt{a}.\sqrt{a}}{\sqrt{a}\left(\sqrt{a}-1\right)}-\frac{1}{\sqrt{a}\left(\sqrt{a}-1\right)}\right):\left(\frac{\sqrt{a}-1}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}+\frac{2}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}\right)=\frac{a-1}{\sqrt{a}\left(\sqrt{a}-1\right)}.\frac{a-1}{\sqrt{a}+1}\)
\(=\frac{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}{\sqrt{a}\left(\sqrt{a}-1\right)}.\frac{a-1}{\sqrt{a}+1}=\frac{a-1}{\sqrt{a}}\)
\(a=2+2\sqrt{2}+1=\left(\sqrt{2}+1\right)^2\). thay a vào rồi tính thôi nha
b) \(K=\frac{a-1}{\sqrt{a}}\). ta thấy \(a>0\Rightarrow\sqrt{a}>0\) => K<0 <=> a-1<0 <=> a<1 => 0<a<1
BT
18 tháng 5 2018
1/ a/ \(K=\left(\frac{\sqrt{a}}{\sqrt{a}-1}-\frac{1}{\sqrt{a}\left(\sqrt{a}-1\right)}\right):\left(\frac{1}{\sqrt{a}+1}+\frac{2}{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}\right)\)
\(K=\frac{a-1}{\sqrt{a}\left(\sqrt{a}-1\right)}:\frac{\sqrt{a}-1+2}{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}\)
\(K=\frac{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}{\sqrt{a}\left(\sqrt{a}-1\right)}:\frac{\sqrt{a}+1}{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}\)
\(K=\frac{\sqrt{a}+1}{\sqrt{a}}:\frac{1}{\sqrt{a}-1}=\frac{\sqrt{a}+1}{\sqrt{a}}.\left(\sqrt{a}-1\right)\)
\(K=\frac{a-1}{\sqrt{a}}\)
b/ Với \(a=3+2\sqrt{2}\) => \(K=\frac{a-1}{\sqrt{a}}=\frac{3+2\sqrt{2}-1}{\sqrt{3+2\sqrt{2}}}=\frac{2+2\sqrt{2}}{\sqrt{2+2\sqrt{2}+1}}=\frac{2\left(\sqrt{2}+1\right)}{\sqrt{\left(\sqrt{2}+1\right)^2}}=\frac{2\left(\sqrt{2}+1\right)}{\left(\sqrt{2}+1\right)}\)
=> \(K=2\)
BT
18 tháng 5 2018
2/ Ta có: x3-y3=x-y)(x2+xy+y2)=(x-y)(x2-2xy+y2+3xy)=(x-y)[(x-y)2+3xy]=9
Thay x-y=3 vào ta được: 3(9+3xy)=9
<=> 3+xy=1 => xy=-2
Ta có hệ PT: \(\hept{\begin{cases}x-y=3\\xy=-2\end{cases}}\)=> \(\hept{\begin{cases}x=y+3\\xy=-2\end{cases}}\)
=> y(y+3)+2=0
<=> y2+3y+2=0
<=> y2+y+2y+2=0 <=> y(y+1)+2(y+1)=0 <=> (y+1)(y+2)=0
=> y1=-1 => x1=2
y2=-2 => x2=1
Đáp số: Các cặp x,y là: (2; -1) và (1; -2)
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