Cho P: \(\left(\dfrac{x-2}{x+2\sqrt{x}}+\dfrac{1}{\sqrt{x}+2}\right).\dfrac{\sqrt{x}+1}{\sqrt{x}-1}\)
CMR: P=\(\dfrac{\sqrt{x}+1}{\sqrt{x}}\)
Những câu hỏi liên quan
left(dfrac{x+2}{xsqrt{x}-1}+dfrac{sqrt{x}}{x+sqrt{x}+1}+dfrac{1}{1-sqrt{x}}right):dfrac{sqrt{x}-1}{2}
left(dfrac{x+2}{left(sqrt{x}-1right)left(x+sqrt{x}+1right)}+dfrac{sqrt{x}left(sqrt{x}-1right)}{left(sqrt{x}-1right)left(x+sqrt{x}+1right)}-dfrac{1}{sqrt{x}-1}right).dfrac{2}{sqrt{x}-1}
dfrac{x+2+x-sqrt{x}-x-sqrt{x}-1}{left(sqrt{x}-1right)left(x+sqrt{x}+1right)}.dfrac{2}{sqrt{x}-1}dfrac{sqrt{x}-1}{x+sqrt{x}+1}.dfrac{2}{sqrt{x}-1}dfrac{2}{x+sqrt{x}+1}
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\(\left(\dfrac{x+2}{x\sqrt{x}-1}+\dfrac{\sqrt{x}}{x+\sqrt{x}+1}+\dfrac{1}{1-\sqrt{x}}\right):\dfrac{\sqrt{x}-1}{2}\)
\(=\left(\dfrac{x+2}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}+\dfrac{\sqrt{x}\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}-\dfrac{1}{\sqrt{x}-1}\right).\dfrac{2}{\sqrt{x}-1}\)
= \(\dfrac{x+2+x-\sqrt{x}-x-\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}.\dfrac{2}{\sqrt{x}-1}\)=\(\dfrac{\sqrt{x}-1}{x+\sqrt{x}+1}.\dfrac{2}{\sqrt{x}-1}=\dfrac{2}{x+\sqrt{x}+1}\)
4.
\(A=\left(\dfrac{\sqrt{x}}{\sqrt{x}-1}\right)-\left(\dfrac{1}{x+\sqrt{x}}\right).\left(\dfrac{1}{\sqrt{x}-1}-\dfrac{2}{x-1}\right)\)
a. Rút gọn A.
b. Tính x khi \(A=\dfrac{1}{2}\)
5. CMR
\(\left(\dfrac{\sqrt{30}}{\sqrt{3}}-\dfrac{\sqrt{20}}{\sqrt{2}}-\dfrac{6}{\sqrt{6}}\right).\sqrt{4+\sqrt{15}}=2\)
nhanh lên nha
9 tháng 8 2021 lúc 17:41
Rút gọn:
1) \(P=\left(\dfrac{\sqrt{x}}{2}-\dfrac{1}{2\sqrt{x}}\right):\left(\dfrac{x-\sqrt{x}}{\sqrt{x}+1}-\dfrac{x-\sqrt{x}}{\sqrt{x}+1}-\dfrac{x+\sqrt{x}}{\sqrt{x}-1}\right)\)
2) \(P=\left(\dfrac{1}{\sqrt{x}}-\dfrac{1}{\sqrt{x}-1}\right):\left(\dfrac{\sqrt{x}+1}{\sqrt{x}-2}-\dfrac{\sqrt{x}+2}{\sqrt{x}-1}\right)\)
Giúp mk nhé :3
Cho biểu thức A:
\(A=\left(\dfrac{x+2}{x\sqrt{x}-1}+\dfrac{\sqrt{x}}{x+1+\sqrt{x}}+\dfrac{1}{1-\sqrt{x}}\right):\dfrac{\sqrt{x}-1}{2\sqrt{x}}\)
a) Rút gọn A.
b) cmr: \(A< \dfrac{2}{3}\)
a) ĐKXĐ: \(x>0,x\ne1\)
\(A=\dfrac{x+2+x-\sqrt{x}-x-\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}.\dfrac{2\sqrt{x}}{\sqrt{x}-1}\)
\(=\dfrac{x-2\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}.\dfrac{2\sqrt{x}}{\sqrt{x}-1}\)
\(=\dfrac{\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}.\dfrac{2\sqrt{x}}{\sqrt{x}-1}=\dfrac{2\sqrt{x}}{x+\sqrt{x}+1}\)
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23 tháng 8 2021 lúc 19:43
Rút gọn:1) Pdfrac{x^2-sqrt{x}}{x+sqrt{x}+1}-dfrac{2x+sqrt{x}}{sqrt{x}}+dfrac{2left(x-1right)}{sqrt{x}-1}2)Aleft(dfrac{2sqrt{x}+x}{xsqrt{x}-1}-dfrac{1}{sqrt{x}-1}right):left(1-dfrac{sqrt{x}+2}{x+sqrt{x}+1}right)3) Aleft(dfrac{1}{sqrt{a}-1}-dfrac{1}{sqrt{a}}right):left(dfrac{sqrt{a}+1}{sqrt{a}-2}-dfrac{sqrt{a}+2}{sqrt{a}-1}right) 4) Aleft(dfrac{1}{sqrt{x}+1}-dfrac{2sqrt{x}-2}{xsqrt{x}-sqrt{x}+x-1}right):left(dfrac{1}{sqrt{x}-1}-dfrac{2}{x-1}right)Mng giúp e vs ạ, cần gấp :
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Rút gọn:
1) \(P=\dfrac{x^2-\sqrt{x}}{x+\sqrt{x}+1}-\dfrac{2x+\sqrt{x}}{\sqrt{x}}+\dfrac{2\left(x-1\right)}{\sqrt{x}-1}\)
2)\(A=\left(\dfrac{2\sqrt{x}+x}{x\sqrt{x}-1}-\dfrac{1}{\sqrt{x}-1}\right):\left(1-\dfrac{\sqrt{x}+2}{x+\sqrt{x}+1}\right)\)
3) \(A=\left(\dfrac{1}{\sqrt{a}-1}-\dfrac{1}{\sqrt{a}}\right):\left(\dfrac{\sqrt{a}+1}{\sqrt{a}-2}-\dfrac{\sqrt{a}+2}{\sqrt{a}-1}\right)\)
4) \(A=\left(\dfrac{1}{\sqrt{x}+1}-\dfrac{2\sqrt{x}-2}{x\sqrt{x}-\sqrt{x}+x-1}\right):\left(\dfrac{1}{\sqrt{x}-1}-\dfrac{2}{x-1}\right)\)
Mng giúp e vs ạ, cần gấp :<
1: Ta có: \(P=\dfrac{x^2-\sqrt{x}}{x+\sqrt{x}+1}-\dfrac{2x+\sqrt{x}}{\sqrt{x}}+\dfrac{2\left(x-1\right)}{\sqrt{x}-1}\)
\(=x-\sqrt{x}-2\sqrt{x}-1+2\sqrt{x}+2\)
\(=x-\sqrt{x}+1\)
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2: Ta có: \(A=\left(\dfrac{x+2\sqrt{x}}{x\sqrt{x}-1}-\dfrac{1}{\sqrt{x}-1}\right):\left(1-\dfrac{\sqrt{x}+2}{x+\sqrt{x}+1}\right)\)
\(=\dfrac{x+2\sqrt{x}-x+\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}:\dfrac{x+\sqrt{x}+1-\sqrt{x}-2}{\left(x+\sqrt{x}+1\right)}\)
\(=\dfrac{\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\cdot\dfrac{x+\sqrt{x}+1}{x-1}\)
\(=\dfrac{1}{x-1}\)
3: Ta có: \(A=\left(\dfrac{1}{\sqrt{a}-1}-\dfrac{1}{\sqrt{a}}\right):\left(\dfrac{\sqrt{a}+1}{\sqrt{a}-2}-\dfrac{\sqrt{a}+2}{\sqrt{a}-1}\right)\)
\(=\dfrac{\sqrt{a}-\sqrt{a}+1}{\sqrt{a}\left(\sqrt{a}-1\right)}:\dfrac{a-1-a+4}{\left(\sqrt{a}-2\right)\left(\sqrt{a}-1\right)}\)
\(=\dfrac{1}{\sqrt{a}\left(\sqrt{a}-1\right)}\cdot\dfrac{\left(\sqrt{a}-2\right)\left(\sqrt{a}-1\right)}{3}\)
\(=\dfrac{\sqrt{a}-2}{3\sqrt{a}}\)
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\(\left(\dfrac{2}{2-\sqrt{x}}+\dfrac{3+\sqrt{x}}{x-2\sqrt{x}}\right):\left(\dfrac{2+\sqrt{x}}{2-\sqrt{x}}-\dfrac{2-\sqrt{x}}{2+\sqrt{x}}-\dfrac{4x}{x-4}\right)\)
\(\left(\dfrac{\sqrt{x}}{x-1}-\dfrac{\sqrt{x}-1}{x+2\sqrt{x}+1}\right).\dfrac{\left(\sqrt{x}+1\right)^2}{3\sqrt{x}-1}\) RÚT GỌN
\(\left(\dfrac{2}{2-\sqrt{x}}+\dfrac{3+\sqrt{x}}{x-2\sqrt{x}}\right):\left(\dfrac{2+\sqrt{x}}{2-\sqrt{x}}-\dfrac{2-\sqrt{x}}{2+\sqrt{x}}-\dfrac{4x}{x-4}\right)\) (ĐK: \(x\ne4;x>0\))
\(=\left[\dfrac{-2\sqrt{x}}{\sqrt{x}\left(\sqrt{x}-2\right)}+\dfrac{\sqrt{x}+3}{\sqrt{x}\left(\sqrt{x}-2\right)}\right]:\left[\dfrac{-\left(\sqrt{x}+2\right)}{\sqrt{x}-2}+\dfrac{\sqrt{x}-2}{\sqrt{x}+2}-\dfrac{4x}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}\right]\)
\(=\dfrac{-2\sqrt{x}+\sqrt{x}+3}{\sqrt{x}\left(\sqrt{x}-2\right)}:\left[\dfrac{-\left(\sqrt{x}+2\right)^2+\left(\sqrt{x}-2\right)^2-4x}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}\right]\)
\(=\dfrac{3-\sqrt{x}}{\sqrt{x}\left(\sqrt{x}-2\right)}:\dfrac{-x-4\sqrt{x}-4+x+4\sqrt{x}+4-4x}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}\)
\(=\dfrac{3-\sqrt{x}}{\sqrt{x}\left(\sqrt{x}-2\right)}:\dfrac{-4x}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}\)
\(=\dfrac{3-\sqrt{x}}{\sqrt{x}\left(\sqrt{x}-2\right)}\cdot\dfrac{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}{-4x}\)
\(=\dfrac{\left(3-\sqrt{x}\right)\left(\sqrt{x}+2\right)}{-4x}\)
\(=-\dfrac{3\sqrt{x}+6-x-2\sqrt{x}}{4x}\)
\(=-\dfrac{\sqrt{x}-x+6}{4x}\)
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\(\left(\dfrac{\sqrt{x}}{x-1}-\dfrac{\sqrt{x}-1}{x+2\sqrt{x}+1}\right)\cdot\dfrac{\left(\sqrt{x}+1\right)^2}{3\sqrt{x}-1}\) (ĐK: \(x\ge0;x\ne1;x\ne\dfrac{1}{9}\))
\(=\left[\dfrac{\sqrt{x}}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}-\dfrac{\sqrt{x}-1}{\left(\sqrt{x}+1\right)^2}\right]\cdot\dfrac{\left(\sqrt{x}+1\right)^2}{3\sqrt{x}-1}\)
\(=\left[\dfrac{\sqrt{x}\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+1\right)^2\left(\sqrt{x}-1\right)}-\dfrac{\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}+1\right)^2\left(\sqrt{x}-1\right)}\right]\cdot\dfrac{\left(\sqrt{x}+1\right)^2}{3\sqrt{x}-1}\)
\(=\dfrac{x+\sqrt{x}-x+2\sqrt{x}-1}{\left(\sqrt{x}+1\right)^2\left(\sqrt{x}-1\right)}\cdot\dfrac{\left(\sqrt{x}+1\right)^2}{3\sqrt{x}-1}\)
\(=\dfrac{3\sqrt{x}-1}{\left(\sqrt{x}+1\right)^2\left(\sqrt{x}-1\right)}\cdot\dfrac{\left(\sqrt{x}+1\right)^2}{3\sqrt{x}-1}\)
\(=\dfrac{1}{\sqrt{x}-1}\)
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Rút gọn:
M1-left[dfrac{2x-1+sqrt{x}}{1-x}+dfrac{2xsqrt{x}+x-sqrt{x}}{1+xsqrt{x}}right]cdotleft[dfrac{left(x-sqrt{x}right)left(1-sqrt{x}right)}{2sqrt{x}-1}right]
Giải::
ĐK: x khác +- 1
M1-left[dfrac{left(sqrt{x}-dfrac{1}{2}right)left(sqrt{x}+1right)}{left(1+sqrt{x}right)left(1-sqrt{x}right)}+dfrac{sqrt{x}left(sqrt{x}-dfrac{1}{2}right)left(sqrt{x}+1right)}{left(1+sqrt{x}right)left(1-sqrt{x}+xright)}right]cdotleft[dfrac{-sqrt{x}left(1-sqrt{x}right)^2}{2left(sqrt{x}-dfrac{1}{2}right)}right...
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Rút gọn:
\(M=1-\left[\dfrac{2x-1+\sqrt{x}}{1-x}+\dfrac{2x\sqrt{x}+x-\sqrt{x}}{1+x\sqrt{x}}\right]\cdot\left[\dfrac{\left(x-\sqrt{x}\right)\left(1-\sqrt{x}\right)}{2\sqrt{x}-1}\right]\)
Giải::
ĐK: x khác +- 1
\(M=1-\left[\dfrac{\left(\sqrt{x}-\dfrac{1}{2}\right)\left(\sqrt{x}+1\right)}{\left(1+\sqrt{x}\right)\left(1-\sqrt{x}\right)}+\dfrac{\sqrt{x}\left(\sqrt{x}-\dfrac{1}{2}\right)\left(\sqrt{x}+1\right)}{\left(1+\sqrt{x}\right)\left(1-\sqrt{x}+x\right)}\right]\cdot\left[\dfrac{-\sqrt{x}\left(1-\sqrt{x}\right)^2}{2\left(\sqrt{x}-\dfrac{1}{2}\right)}\right]\)
\(=1-\left[\dfrac{\left(\sqrt{x}-\dfrac{1}{2}\right)}{\left(1-\sqrt{x}\right)}\cdot\dfrac{-\sqrt{x}\left(1-\sqrt{x}\right)^2}{2\left(\sqrt{x}-\dfrac{1}{2}\right)}+\dfrac{\sqrt{x}\left(\sqrt{x}-\dfrac{1}{2}\right)}{1-\sqrt{x}+x}\cdot\dfrac{-\sqrt{x}\left(1-\sqrt{x}\right)^2}{2\left(\sqrt{x}-\dfrac{1}{2}\right)}\right]\)
\(=1-\left[\dfrac{-\sqrt{x}\left(1-\sqrt{x}\right)}{2}+\dfrac{-x\left(1-\sqrt{x}\right)^2}{2\left(1-\sqrt{x}+x\right)}\right]\)
rồi làm sao nữa ak?? Tớ có quy đồng lên, tính sơ sơ rồi nhưng thấy kq không gọn.
Câu b là : tìm các số nguyên x để M cũng là số nguyên . Nên tớ nghĩ kq sẽ gọn.
NHỜ MẤY CAO NHÂN RA TAY GIÚP VỚI NHAK ^^!
1.Aleft(dfrac{x+2}{xsqrt{x}-1}+dfrac{sqrt{x}}{x+sqrt{x}+1}+dfrac{1}{1-sqrt{x}}right):dfrac{sqrt{x}}{2}2.Bleft(dfrac{2sqrt{x+x+1}}{sqrt{x}+1}right)left(1-dfrac{x-sqrt{x}}{sqrt{x}-1}right):left(1-sqrt{x}right)3.Cleft(dfrac{sqrt{x}+1}{sqrt{x}-1}-dfrac{sqrt{x}-1}{sqrt{x}+1}dfrac{8sqrt{x}}{x-1}right):left(dfrac{sqrt{x}-x-3}{x-1}-dfrac{1}{sqrt{x}-1}right)Làm chi tiết hộ mình với ak mình đang cần gấp!!!
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1.A=\(\left(\dfrac{x+2}{x\sqrt{x}-1}+\dfrac{\sqrt{x}}{x+\sqrt{x}+1}+\dfrac{1}{1-\sqrt{x}}\right):\dfrac{\sqrt{x}}{2}\)
2.B=\(\left(\dfrac{2\sqrt{x+x+1}}{\sqrt{x}+1}\right)\left(1-\dfrac{x-\sqrt{x}}{\sqrt{x}-1}\right):\left(1-\sqrt{x}\right)\)
3.C=\(\left(\dfrac{\sqrt{x}+1}{\sqrt{x}-1}-\dfrac{\sqrt{x}-1}{\sqrt{x}+1}\dfrac{8\sqrt{x}}{x-1}\right):\left(\dfrac{\sqrt{x}-x-3}{x-1}-\dfrac{1}{\sqrt{x}-1}\right)\)
Làm chi tiết hộ mình với ak mình đang cần gấp!!!
1: Ta có: \(A=\left(\dfrac{x+2}{x\sqrt{x}-1}+\dfrac{\sqrt{x}}{x+\sqrt{x}+1}+\dfrac{1}{1-\sqrt{x}}\right):\dfrac{\sqrt{x}}{2}\)
\(=\dfrac{x+2+x-\sqrt{x}-x-\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\cdot\dfrac{2}{\sqrt{x}}\)
\(=\dfrac{x-2\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\cdot\dfrac{2}{\sqrt{x}}\)
\(=\dfrac{2\sqrt{x}-2}{\sqrt{x}\left(x+\sqrt{x}+1\right)}\)
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Cho 3 số dương x,y,z. CMR:\(\dfrac{1}{\sqrt{x}}+\dfrac{1}{\sqrt{y}}+\dfrac{1}{\sqrt{z}}>=3\left(\dfrac{1}{\sqrt{x}+2\sqrt{y}}+\dfrac{1}{\sqrt{y}+2\sqrt{z}}+\dfrac{1}{\sqrt{z}+2\sqrt{x}}\right)\)
Áp dụng BĐT Cauchy-Schwarz ta có:
\(\dfrac{1}{\sqrt{x}+2\sqrt{y}}\le\dfrac{1}{9}\left(\dfrac{1}{\sqrt{x}}+\dfrac{1}{\sqrt{y}}+\dfrac{1}{\sqrt{y}}\right)\)
Tương tự cho 2 BĐT trên ta có:
\(\dfrac{1}{3}VP\le\dfrac{1}{9}\cdot3\left(\dfrac{1}{\sqrt{x}}+\dfrac{1}{\sqrt{y}}+\dfrac{1}{\sqrt{z}}\right)\)
\(=\dfrac{1}{3}\left(\dfrac{1}{\sqrt{x}}+\dfrac{1}{\sqrt{y}}+\dfrac{1}{\sqrt{z}}\right)=\dfrac{1}{3}VT\)
Xảy ra khi \(x=y=z\)
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