Rút gọn:
\(A=1+\sqrt{x-1}+\sqrt{x-2\sqrt{x-1}}\)
Những câu hỏi liên quan
2 a. rút gọn biểu C = \(\dfrac{2x^{\text{2}}-x}{\text{x }-1}+\dfrac{x+1}{1-x}+\dfrac{2-x^2}{x-1}\)
b. Rút gọn biểu thức D = \(\left(\dfrac{1}{a-\sqrt{a}}+\dfrac{1}{\sqrt{\text{a}}-1}\right):\dfrac{\sqrt{\text{a}}+1}{a-2\sqrt{a}+1}\)
Vậy khi rút gọn một biểu thức hửu tỉ và một biểu thức chứa căn có tìm điều kiện xác định không?
\(a,C=\dfrac{2x^2-x-x-1+2-x^2}{x-1}\left(x\ne1\right)\\ C=\dfrac{x^2-2x+1}{x-1}=\dfrac{\left(x-1\right)^2}{x-1}=x-1\\ b,D=\dfrac{1+\sqrt{a}}{\sqrt{a}\left(\sqrt{a}-1\right)}\cdot\dfrac{\left(\sqrt{a}-1\right)^2}{\sqrt{a}+1}\left(a>0;a\ne1\right)\\ D=\dfrac{\sqrt{a}-1}{\sqrt{a}}\)
Có
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A=\(\sqrt{x-2\sqrt{x-1}}+\sqrt{x+2\sqrt{x-1}}\) với x>=1 rút gọn A
Điều kiện: \(x\ge1\)
\(A=\sqrt{x-2\sqrt{x-1}}+\sqrt{x+2\sqrt{x-1}}\\ A=\sqrt{x-1-2\sqrt{x-1}+1}+\sqrt{x-1+2\sqrt{x-1}+1}\\ A=\sqrt{\left(\sqrt{x-1}-1\right)^2}+\sqrt{\left(\sqrt{x-1}+1\right)^2}\\ A=\left|\sqrt{x-1}-1\right|+\sqrt{x-1}+1\)
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1) Rút gọn biểu thứ
A=\(\left(\dfrac{x-y}{\sqrt{x}-\sqrt{y}}+\dfrac{\sqrt{x^3}-\sqrt{y^3}}{y-x}\right):\dfrac{\left(\sqrt{x}-\sqrt{y}\right)^2+\sqrt{xy}}{\sqrt{x}+\sqrt{y}}\)
a) Rút gọn A
b) Chứng minh A<1
Lời giải:
a) ĐK: $x\geq 0; y\geq 0; x\neq y$
\(A=\left[\frac{(\sqrt{x}-\sqrt{y})(\sqrt{x}+\sqrt{y})}{\sqrt{x}-\sqrt{y}}-\frac{(\sqrt{x}-\sqrt{y})(x+\sqrt{xy}+y)}{(\sqrt{x}-\sqrt{y})(\sqrt{x}+\sqrt{y})}\right]:\frac{x-\sqrt{xy}+y}{\sqrt{x}+\sqrt{y}}\)
\(=\left(\sqrt{x}+\sqrt{y}-\frac{x+\sqrt{xy}+y}{\sqrt{x}+\sqrt{y}}\right).\frac{\sqrt{x}+\sqrt{y}}{x-\sqrt{xy}+y}\)
\(=\frac{\sqrt{xy}}{\sqrt{x}+\sqrt{y}}.\frac{\sqrt{x}+\sqrt{y}}{x-\sqrt{xy}+y}=\frac{\sqrt{xy}}{x-\sqrt{xy}+y}\)
b) \(1-A=\frac{(\sqrt{x}-\sqrt{y})^2}{x-\sqrt{xy}+y}>0\) với mọi $x\neq y; x,y\geq 0$
$\Rightarrow A< 1$
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14 tháng 8 2021 lúc 8:33
Rút gọn : A=\(\dfrac{\sqrt{x+2\sqrt{x-1}}+\sqrt{x-2\sqrt{x-1}}}{\sqrt{x+\sqrt{2x-1}+\sqrt{x-\sqrt{2x-1}}}}.\left(\sqrt{2x-1}\right)\)
\(A=\dfrac{2x+2\sqrt{x}+2}{x-1}-\dfrac{\sqrt{x}-1}{\sqrt{x}+1}\)
Rút gọn \(A=\dfrac{\sqrt{x}+3}{\sqrt{x}-1}\)
\(\Leftrightarrow A=\dfrac{2x+2\sqrt{x}+2}{\sqrt{x}^2-1}-\dfrac{\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\dfrac{2x+2\sqrt{x}+2-\left(\sqrt{x}^2+1-2\sqrt{x}\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\\ =\dfrac{2x+2\sqrt{x}+2-x-1+2\sqrt{x}}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\\ =\dfrac{x+4\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\\=\dfrac{x+4\sqrt{x}+1}{x-1} \)
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rút gọn A=((3x+\sqrt(9x)-3)/(x+\sqrt(x)-2)+(1)/(\sqrt(x)-1)+(1)/(\sqrt(x)+2)):(1)/(x-1)
Cho A=((1)/(x-1)+(3\sqrt(x)+5)/(x\sqrt(s)-x-\sqrt(x)+1))*[(|\sqrt(x)+1|^(2))/(4\sqrt(x))-1]a) Rút gọn A
A=\(\dfrac{x+2}{x\sqrt{x}}+\dfrac{\sqrt{x}}{x+\sqrt{x}+1}+\dfrac{1}{1-\sqrt{x}}\)
Rút gọn A
Cho A=\(\left(\dfrac{\sqrt{x}+2}{x+2\sqrt{x}-1}-\dfrac{\sqrt{x}-2}{x-1}+4\sqrt{x}\right):\left(\dfrac{\sqrt{x}}{\sqrt{x+1}}\right)\)
Rút gọn A
Sửa đề: \(A=\left(\dfrac{\sqrt{x}+2}{x+2\sqrt{x}+1}-\dfrac{\sqrt{x}-2}{x-1}+4\sqrt{x}\right):\dfrac{\sqrt{x}}{\sqrt{x}+1}\)
\(=\dfrac{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)-\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)+4\sqrt{x}\left(x-1\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+1\right)^2\left(\sqrt{x}-1\right)}\cdot\dfrac{\sqrt{x}+1}{\sqrt{x}}\)
\(=\dfrac{x+\sqrt{x}-2-x+\sqrt{x}+2+4\sqrt{x}\left(\sqrt{x}+1\right)^2\left(\sqrt{x}-1\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)\sqrt{x}}\)
\(=\dfrac{2\sqrt{x}\left[1+2\left(x+2\sqrt{x}+1\right)\left(\sqrt{x}-1\right)\right]}{\sqrt{x}\left(x-1\right)}\)
\(=\dfrac{2\left[1+2\left(x\sqrt{x}-x+2x-2\sqrt{x}+\sqrt{x}-1\right)\right]}{x-1}\)
\(=\dfrac{2\left[1+2x\sqrt{x}+2x-2\sqrt{x}-2\right]}{x-1}=\dfrac{2\left(2x\sqrt{x}+2x-2\sqrt{x}-1\right)}{x-1}\)
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Rút gọn A=\(\left(\dfrac{2\sqrt{x}+x}{x\sqrt{x}-1}-\dfrac{1}{\sqrt{x}-1}\right):\left(1-\dfrac{\sqrt{x}+2}{x+\sqrt{x}+1}\right)\)
\(=\dfrac{x+2\sqrt{x}-x-\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}:\dfrac{x+\sqrt{x}+1-\sqrt{x}-2}{x+\sqrt{x}+1}\)
\(=\dfrac{\sqrt{x}-1}{\left(\sqrt{x}-1\right)}\cdot\dfrac{1}{x-1}=\dfrac{1}{x-1}\)
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