Cho A=\(\frac{\sqrt{x}}{\sqrt{x}+1}+\frac{3}{\sqrt{x}+1}-\frac{6\sqrt{x}-4}{x-1}\left(x\ge0;x\ne1\right)\)
Tìm GTNN của A
Những câu hỏi liên quan
Rút gọn biểu thức:
a) Aleft(frac{3x-3sqrt{x}-3}{x+sqrt{x}-2}+frac{1}{sqrt{x}-1}-frac{1}{sqrt{x}+2}right):frac{1}{sqrt{x}+2}left(xge0,xne1right)
b) Bfrac{xsqrt{x}-3}{x-2sqrt{x}-3}-frac{2left(sqrt{x-3}right)}{sqrt{x}+1}+frac{sqrt{x}+3}{3-sqrt{x}}left(x0,xne9right)
c) Cfrac{2sqrt{x}-9}{x-5+6}-frac{sqrt{x}+3}{sqrt{x}-2}-frac{2sqrt{x}+1}{3-sqrt{x}}left(xge0,xne4,xne9right)
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Rút gọn biểu thức:
a) \(A=\left(\frac{3x-3\sqrt{x}-3}{x+\sqrt{x}-2}+\frac{1}{\sqrt{x}-1}-\frac{1}{\sqrt{x}+2}\right):\frac{1}{\sqrt{x}+2}\left(x\ge0,x\ne1\right)\)
b) \(B=\frac{x\sqrt{x}-3}{x-2\sqrt{x}-3}-\frac{2\left(\sqrt{x-3}\right)}{\sqrt{x}+1}+\frac{\sqrt{x}+3}{3-\sqrt{x}}\left(x>0,x\ne9\right)\)
c) \(C=\frac{2\sqrt{x}-9}{x-5+6}-\frac{\sqrt{x}+3}{\sqrt{x}-2}-\frac{2\sqrt{x}+1}{3-\sqrt{x}}\left(x\ge0,x\ne4,x\ne9\right)\)
1. sqrt{x-2sqrt{x-1}}+sqrt{x+3-4sqrt{x-1}}left(2 x 5right)2. frac{6}{1-sqrt{3}}-frac{3sqrt{3}-1}{sqrt{3}+1}+sqrt{3}3. sqrt{29-12sqrt{5}+sqrt{24-8sqrt{3}}}4. sqrt{frac{4}{9-4sqrt{5}}}-sqrt{frac{4}{9+4sqrt{5}}}5. 5sqrt{frac{1}{5}}+frac{1}{2}sqrt{x}-frac{5}{4}sqrt{frac{4}{5}+sqrt{5}}6. frac{6-sqrt{6}}{sqrt{6}-1}-9sqrt{frac{2}{3}}-frac{4}{2-sqrt{6}}7. left(frac{sqrt{x}-2}{x-1}-frac{sqrt{x}+2}{x+2sqrt{x}+1}right)frac{left(sqrt{x}-1right)^2}{2}left(xge0,xne1right)
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1. \(\sqrt{x-2\sqrt{x-1}}+\sqrt{x+3-4\sqrt{x-1}}\left(2< x< 5\right)\)
2. \(\frac{6}{1-\sqrt{3}}-\frac{3\sqrt{3}-1}{\sqrt{3}+1}+\sqrt{3}\)
3. \(\sqrt{29-12\sqrt{5}+\sqrt{24-8\sqrt{3}}}\)
4. \(\sqrt{\frac{4}{9-4\sqrt{5}}}-\sqrt{\frac{4}{9+4\sqrt{5}}}\)
5. \(5\sqrt{\frac{1}{5}}+\frac{1}{2}\sqrt{x}-\frac{5}{4}\sqrt{\frac{4}{5}+\sqrt{5}}\)
6. \(\frac{6-\sqrt{6}}{\sqrt{6}-1}-9\sqrt{\frac{2}{3}}-\frac{4}{2-\sqrt{6}}\)
7. \(\left(\frac{\sqrt{x}-2}{x-1}-\frac{\sqrt{x}+2}{x+2\sqrt{x}+1}\right)\frac{\left(\sqrt{x}-1\right)^2}{2}\left(x\ge0,x\ne1\right)\)
Trả lời nhanh giúp mình với mình cần gấp lắm
cho biểu thức: Pleft(frac{sqrt{x}+1}{sqrt{xy}+1}+frac{sqrt{xy}+sqrt{x}}{1-sqrt{xy}}+1right):left(1-frac{sqrt{xy}+sqrt{x}}{sqrt{xy}-1}-frac{sqrt{x}+1}{sqrt{xy}+1}right) Pleft(frac{sqrt{x}+1}{sqrt{xy}+1}+frac{sqrt{xy}+sqrt{x}}{1-sqrt{xy}}+1right):left(1-frac{sqrt{xy}+1}{sqrt{xy}-1}-frac{sqrt{x}+1}{sqrt{xy}+1}right).backslash với x,yge0;x,yne1 a) Rút gọn Pb) Tính P khi xsqrt[3]{4-2sqrt{6}}+sqrt[3]{4+2sqrt{6}}và yx^2+6
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cho biểu thức: \(P=\left(\frac{\sqrt{x}+1}{\sqrt{xy}+1}+\frac{\sqrt{xy}+\sqrt{x}}{1-\sqrt{xy}}+1\right):\left(1-\frac{\sqrt{xy}+\sqrt{x}}{\sqrt{xy}-1}-\frac{\sqrt{x}+1}{\sqrt{xy}+1}\right)\) \(P=\left(\frac{\sqrt{x}+1}{\sqrt{xy}+1}+\frac{\sqrt{xy}+\sqrt{x}}{1-\sqrt{xy}}+1\right):\left(1-\frac{\sqrt{xy}+1}{\sqrt{xy}-1}-\frac{\sqrt{x}+1}{\sqrt{xy}+1}\right).\backslash\ \)với \(x,y\ge0;x,y\ne1\)
a) Rút gọn P
b) Tính P khi \(x=\sqrt[3]{4-2\sqrt{6}}+\sqrt[3]{4+2\sqrt{6}}\)và \(y=x^2+6\)
a/ \(P=\frac{1}{\sqrt{xy}}\)
b/ \(x^3=8-6x\)
\(\Rightarrow P=\frac{1}{\sqrt{x\left(x^2+6\right)}}=\frac{1}{\sqrt{x^3+6x}}=\frac{1}{\sqrt{8-6x+6x}}=\frac{1}{2\sqrt{2}}\)
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Rút gọn các biểu thức sau:
a) A=\(\frac{1}{\sqrt{3}+1}+\frac{1}{\sqrt{3}-1}+\frac{2\sqrt{2}-\sqrt{6}}{\sqrt{2}}\)
b)B=\(\left(\frac{1}{x-\sqrt{x}}+\frac{1}{\sqrt{x}-1}\right):\frac{\sqrt{x}}{x-2\sqrt{x}+1}\left(x\ge0;x\ne1\right)\)
Lời giải:
a)
\(A=\frac{\sqrt{3}-1+\sqrt{3}+1}{(\sqrt{3}+1)(\sqrt{3}-1)}+2-\sqrt{3}=\frac{2\sqrt{3}}{3-1}+2-\sqrt{3}=\sqrt{3}+2-\sqrt{3}=2\)
b)
\(B=\left(\frac{1}{\sqrt{x}(\sqrt{x}-1)}+\frac{\sqrt{x}}{\sqrt{x}(\sqrt{x}-1)}\right):\frac{\sqrt{x}}{(\sqrt{x}-1)^2}\)
\(=\frac{\sqrt{x}+1}{\sqrt{x}.(\sqrt{x}-1)}.\frac{(\sqrt{x}-1)^2}{\sqrt{x}}=\frac{(\sqrt{x}-1)(\sqrt{x}+1)}{x}=\frac{x-1}{x}\)
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Chứng minh các đẳng thức saua) left(frac{2sqrt{6}-sqrt{3}}{2sqrt{2}-1}+frac{5+2sqrt{5}}{2+sqrt{5}}right)left(sqrt{5}-sqrt{3}right)b) frac{a-b}{b^2}sqrt{frac{a^2b^4}{a^2-2ab+b^2}}-a(Với ba0c)left(sqrt{a}+frac{1-asqrt{a}}{1-sqrt{a}}right)left(frac{1-sqrt{a}}{1-a}right)^21với age0,a khác 1d) left(frac{3sqrt{5}-sqrt{15}}{sqrt{27}-3}+frac{2sqrt{5}}{sqrt{3}}right)40sqrt{15}600e) left(1+frac{x+sqrt{x}}{sqrt{x}+1}right)left(1-frac{x-sqrt{x}}{sqrt{x}-1}right)1-xvới xge0;xne1
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Chứng minh các đẳng thức sau
a) \(\left(\frac{2\sqrt{6}-\sqrt{3}}{2\sqrt{2}-1}+\frac{5+2\sqrt{5}}{2+\sqrt{5}}\right)\left(\sqrt{5}-\sqrt{3}\right)\)
b) \(\frac{a-b}{b^2}\sqrt{\frac{a^2b^4}{a^2-2ab+b^2}}=-a\)(Với b<a<0
c)\(\left(\sqrt{a}+\frac{1-a\sqrt{a}}{1-\sqrt{a}}\right)\left(\frac{1-\sqrt{a}}{1-a}\right)^2=1\)với a\(\ge0\),a khác 1
d) \(\left(\frac{3\sqrt{5}-\sqrt{15}}{\sqrt{27}-3}+\frac{2\sqrt{5}}{\sqrt{3}}\right)40\sqrt{15}=600\)
e) \(\left(1+\frac{x+\sqrt{x}}{\sqrt{x}+1}\right)\left(1-\frac{x-\sqrt{x}}{\sqrt{x}-1}\right)=1-x\)với x\(\ge0;x\ne1\)
Rút gọn:
a) \(A=\left(\frac{1-x\sqrt{x}}{1-\sqrt{x}}+\sqrt{x}\right)\left(\frac{1-\sqrt{x}}{1-x}\right)^2\left(x\ge0,x\ne1\right)\)
b) \(B=\left(\frac{2-a\sqrt{a}}{2-\sqrt{a}}+\sqrt{a}\right)\left(\frac{2-\sqrt{a}}{2-a}\right)\left(a\ge0,a\ne2,a\ne4\right)\)
c) \(C=\frac{x\sqrt{x}-1}{x-\sqrt{x}}-\frac{x\sqrt{x}+1}{x+\sqrt{x}}+\frac{x+1}{\sqrt{x}}\left(x>0,x\ne1\right)\)
a) Ta có: \(A=\left(\frac{1-x\sqrt{x}}{1-\sqrt{x}}+\sqrt{x}\right)\cdot\left(\frac{1-\sqrt{x}}{1-x}\right)^2\)
\(=\left(\frac{1-x\sqrt{x}+\sqrt{x}\left(1-\sqrt{x}\right)}{1-\sqrt{x}}\right)\cdot\left(\frac{1}{1+\sqrt{x}}\right)^2\)
\(=\frac{1-x\sqrt{x}+\sqrt{x}-x}{1-\sqrt{x}}\cdot\frac{1}{\left(1+\sqrt{x}\right)^2}\)
\(=\frac{-\left(x-1\right)\left(-1-\sqrt{x}\right)}{1-\sqrt{x}}\cdot\frac{1}{\left(1+\sqrt{x}\right)^2}\)
\(=\frac{\left(1+\sqrt{x}\right)\cdot\left(-1-\sqrt{x}\right)}{\left(1+\sqrt{x}\right)^2}\)
\(=\frac{-1\cdot\left(1+\sqrt{x}\right)^2}{\left(1+\sqrt{x}\right)^2}=-1\)
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A = \(\frac{\left(1+\sqrt{x}\right)^2-4\sqrt{x}}{\sqrt{x}-1}\left(x\ge0\right)\left(x\ne1\right)\)
B = \(\frac{3+2\sqrt{3}}{\sqrt{3}}+\frac{1}{\sqrt{3}-\sqrt{2}}+\frac{2+\sqrt{2}}{\sqrt{x}+1}\)
Tìm các giá trị của x để A = B
\(A=\frac{\left(1+\sqrt{x}\right)^2-4\sqrt{x}}{\sqrt{x}-1}\) \(\left(x\ge0;x\ne1\right)\)
\(A=\frac{x+2\sqrt{x}+1-4\sqrt{x}}{\sqrt{x}-1}=\frac{x-2\sqrt{x}+1}{\sqrt{x}-1}=\frac{\left(\sqrt{x}-1\right)^2}{\sqrt{x}-1}=\sqrt{x}-1\)
và \(B=\frac{3+2\sqrt{3}}{\sqrt{3}}+\frac{1}{\sqrt{3}-\sqrt{2}}+\frac{2+\sqrt{2}}{\sqrt{x}+1}\)
\(B=\frac{\sqrt{3}\left(\sqrt{3}+2\right)}{\sqrt{3}}+\frac{1}{\sqrt{3}-\sqrt{2}}+\frac{\sqrt{2}\left(\sqrt{2}+1\right)}{\sqrt{2}+1}\)
\(B=\sqrt{3}+2+\frac{1}{\sqrt{3}-\sqrt{2}}+\sqrt{2}\)
\(B=\sqrt{3}+\sqrt{2}+\frac{1}{\sqrt{3}-\sqrt{2}}+2\)
\(B=\frac{\left(\sqrt{3}+\sqrt{2}\right)\left(\sqrt{3}-\sqrt{2}\right)+1}{\sqrt{3}-\sqrt{2}}+2\)
\(B=\frac{3-2+1}{\sqrt{3}-\sqrt{2}}+2\)
\(B=\frac{2}{\sqrt{3}-\sqrt{2}}+2\)
để A = B thì \(\sqrt{x}-1\)= \(\frac{2}{\sqrt{3}-\sqrt{2}}+2\)
\(\sqrt{x}=\frac{2}{\sqrt{3}-\sqrt{2}}+3\)
\(\sqrt{x}=\frac{2\left(\sqrt{3}+\sqrt{2}\right)}{\left(\sqrt{3}+\sqrt{2}\right)\left(\sqrt{3}-\sqrt{2}\right)}+3\)
\(\sqrt{x}=2\sqrt{3}+2\sqrt{2}+3\)
tới bước này tui bí :(( mong các bạn giỏi khác giúp bạn :D
Đề bài: chứng minh đẳng thức:a) frac{sqrt{a}}{sqrt{a}-sqrt{b}}-frac{sqrt{b}}{sqrt{a}+sqrt{b}}-frac{2b}{a-b}1với a0,b0,ane bleft(1+frac{a+sqrt{a}}{1+sqrt{a}}right).left(1-frac{a-sqrt{a}}{sqrt{a}-1}right)+a1với ane1,age0c) frac{2sqrt{x}-9}{x-5sqrt{x}+6}-frac{sqrt{x}+3}{sqrt{x}-2}-frac{2sqrt{x}+1}{3-sqrt{x}}frac{sqrt{x}+1}{sqrt{x}+3}với xge0,xne4,xne9d) left(frac{x+1}{x^3+1}-frac{1}{-x^2+x-1}-frac{2}{x+1}right):frac{x^3-2x^2}{x^3-x^2+x}+1frac{x-1}{x+1}vớixne0,xne-1,xne2
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Đề bài: chứng minh đẳng thức:
a) \(\frac{\sqrt{a}}{\sqrt{a}-\sqrt{b}}-\frac{\sqrt{b}}{\sqrt{a}+\sqrt{b}}-\frac{2b}{a-b}=1\)với \(a>0,b>0,a\ne b\)
\(\left(1+\frac{a+\sqrt{a}}{1+\sqrt{a}}\right).\left(1-\frac{a-\sqrt{a}}{\sqrt{a}-1}\right)+a=1\)với \(a\ne1,a\ge0\)
c) \(\frac{2\sqrt{x}-9}{x-5\sqrt{x}+6}-\frac{\sqrt{x}+3}{\sqrt{x}-2}-\frac{2\sqrt{x}+1}{3-\sqrt{x}}=\frac{\sqrt{x}+1}{\sqrt{x}+3}\)với \(x\ge0,x\ne4,x\ne9\)
d) \(\left(\frac{x+1}{x^3+1}-\frac{1}{-x^2+x-1}-\frac{2}{x+1}\right):\frac{x^3-2x^2}{x^3-x^2+x}+1=\frac{x-1}{x+1}\)với\(x\ne0,x\ne-1,x\ne2\)
Mới đc câu a ak, thog cảm nha, trih độ mih thấp lắm:
\(\frac{\sqrt{a}}{\sqrt{a}-\sqrt{b}}-\frac{\sqrt{b}}{\sqrt{a}+\sqrt{b}}-\frac{2b}{a-b}\)
=\(\frac{a+\sqrt{ab}-\sqrt{ab}+b}{a-b}-\frac{2b}{a-b}\)
=\(\frac{a+b-2b}{a-b}=\frac{a-b}{a-b}=1\)
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\(\left(1+\frac{a+\sqrt{a}}{1+\sqrt{a}}\right)\left(1-\frac{a-\sqrt{a}}{\sqrt{a}-1}\right)+a\)=\(1-\frac{a-\sqrt{a}}{\sqrt{a}-1}+\frac{a+\sqrt{a}}{1+\sqrt{a}}-\frac{\left(a+\sqrt{a}\right)\left(a-\sqrt{a}\right)}{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}+a\)
=\(1-\frac{\sqrt{a}\left(\sqrt{a}-1\right)}{\sqrt{a}-1}+\frac{\sqrt{a}\left(\sqrt{a}+1\right)}{\sqrt{a}+1}-\frac{a\left(a-1\right)}{a-1}+a\)=\(1-\sqrt{a}+\sqrt{a}-a+a=1\)
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Xem thêm câu trả lời
Cho biểu thức:\(A=1+\left(\frac{2x+\sqrt{x}-1}{1-x}-\frac{2x\sqrt{x}-\sqrt{x}+x}{1-x\sqrt{x}}\right)\frac{x-\sqrt{x}}{2\sqrt{x}-1}\)
a) tìm các giá trị của x để \(A=\frac{6-\sqrt{6}}{5}\)
b)chứng minh rằng \(A>\frac{2}{3}\)với mọi x thỏa mãn \(x\ge0,x\ne1,x\ne\frac{1}{4}\)
a) ĐK: \(x\ge0,x\ne1,x\ne\frac{1}{4}\)
\(A=1+\left(\frac{2x+\sqrt{x}-1}{1-x}-\frac{2x\sqrt{x}-\sqrt{x}+x}{1-x\sqrt{x}}\right)\frac{x-\sqrt{x}}{2\sqrt{x}-1}\)
\(A=1+\left[\frac{\left(2\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+1\right)\left(1-\sqrt{x}\right)}-\frac{\sqrt{x}\left(2\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\left(1-\sqrt{x}\right)\left(x+\sqrt{x}+1\right)}\right]\frac{\sqrt{x}\left(\sqrt{x}-1\right)}{2\sqrt{x}-1}\)
\(A=1+\left[\frac{2\sqrt{x}-1}{1-\sqrt{x}}-\frac{\sqrt{x}\left(2\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\left(1-\sqrt{x}\right)\left(x+\sqrt{x}+1\right)}\right]\frac{\sqrt{x}\left(\sqrt{x}-1\right)}{2\sqrt{x}-1}\)
\(A=1-\sqrt{x}+\frac{x\left(\sqrt{x}+1\right)}{x+\sqrt{x}+1}\)
\(A=\frac{x+1}{x+\sqrt{x}+1}\)
Để \(A=\frac{6-\sqrt{6}}{5}\Rightarrow\frac{x+1}{x+\sqrt{x}+1}=\frac{6-\sqrt{6}}{5}\)
\(\Rightarrow5x+5=\left(6-\sqrt{6}\right)x+\left(6-\sqrt{6}\right)\sqrt{x}+6-\sqrt{6}\)
\(\Rightarrow\left(1-\sqrt{6}\right)x+\left(6-\sqrt{6}\right)\sqrt{x}+1-\sqrt{6}=0\)
\(\Rightarrow x-\sqrt{6}.\sqrt{x}+1=0\)
\(\Rightarrow\orbr{\begin{cases}\sqrt{x}=\frac{\sqrt{2}+\sqrt{6}}{2}\\\sqrt{x}=\frac{-\sqrt{2}+\sqrt{6}}{2}\end{cases}}\Rightarrow\orbr{\begin{cases}x=2+\sqrt{3}\\x=2-\sqrt{3}\end{cases}}\left(tmđk\right)\)
b) Xét \(A-\frac{2}{3}=\frac{x+1}{x+\sqrt{x}+1}-\frac{2}{3}=\frac{3x+3-2x-2\sqrt{x}-2}{3\left(x+\sqrt{x}+1\right)}\)
\(=\frac{x-2\sqrt{x}+1}{3\left(x+\sqrt{x}+1\right)}=\frac{\left(\sqrt{x}-1\right)^2}{3\left(x+\sqrt{x}+1\right)}\)
Do \(x\ge0,x\ne1,x\ne\frac{1}{4}\Rightarrow\left(\sqrt{x}-1\right)^2>0\)
Lại có \(x+\sqrt{x}+1=\left(\sqrt{x}+\frac{1}{2}\right)+\frac{3}{4}>0\)
Nên \(A-\frac{2}{3}>0\Rightarrow A>\frac{2}{3}\).
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