Chứng minh:
\(\left(\frac{\sqrt{x}+\sqrt{y}}{\sqrt{x}-\sqrt{y}}-\frac{\sqrt{x}-\sqrt{y}}{\sqrt{x}+\sqrt{y}}\right):\frac{\sqrt{xy}}{x-y}=4\)
Những câu hỏi liên quan
Chứng minh biểu thức sau không phụ thuộc vào biến:
\(\left(\frac{2\sqrt{xy}}{x-y} +\frac{\sqrt{x}-\sqrt{y}}{2\left(\sqrt{x}+\sqrt{y}\right)}\right).\frac{2\sqrt{x}}{\sqrt{x}+\sqrt{y}}+\frac{y}{\sqrt{y}-\sqrt{x}}\)với x>0 ; y>0 ; x # y
Rút gọn
\(A=\left(\sqrt{ab}-\frac{ab}{a+\sqrt{ab}}\right):\frac{\sqrt[4]{ab}-\sqrt{b}}{a-b}\)
\(B=\left(\sqrt{x}+\frac{y-\sqrt{xy}}{\sqrt{x}+\sqrt{y}}\right):\left(\frac{x}{\sqrt{xy}+y}+\frac{y}{\sqrt{xy}+x}-\frac{x+y}{\sqrt{xy}}\right)\)
Rút gọn
\(A=\left(\sqrt{ab}-\frac{ab}{a+\sqrt{ab}}\right):\frac{\sqrt[4]{ab}-\sqrt{b}}{a-b}\)
\(B=\left(\sqrt{x}+\frac{y-\sqrt{xy}}{\sqrt{x}+\sqrt{y}}\right):\left(\frac{x}{\sqrt{xy}+y}+\frac{y}{\sqrt{xy}-y}-\frac{x-y}{\sqrt{xy}}\right)\)
Cho biểu thức
A= \(\left(\frac{x-y}{\sqrt{x}-\sqrt{y}}+\frac{\sqrt{x^3-\sqrt{y^3}}}{y-x}\right):\frac{\left(\sqrt{x}-\sqrt{y}\right)^2+\sqrt{xy}}{\sqrt{x}+\sqrt{y}}\)
a, Rút gọn A
Chứng minh A>0
\(\left(\sqrt{x}+\frac{y+\sqrt{xy}}{\sqrt{x}+\sqrt{y}}\right):\left(\frac{x}{\sqrt{xy}+y}+\frac{y}{\sqrt{xy}+x}-\frac{x+y}{\sqrt{xy}}\right)\)
\(\left[\sqrt{x}+\frac{y-\sqrt{xy}}{\sqrt{x}+\sqrt{y}}\right]:\left[\frac{x}{\sqrt{xy}+y}+\frac{y}{\sqrt{xy}-x}-\frac{x+y}{\sqrt{xy}}\right]\)
Ta có: \(\left(\sqrt{x}+\frac{y-\sqrt{xy}}{\sqrt{x}+\sqrt{y}}\right):\left(\frac{x}{\sqrt{xy}+y}+\frac{y}{\sqrt{xy}-x}-\frac{x+y}{\sqrt{xy}}\right)\)
\(=\frac{x+\sqrt{xy}+y-\sqrt{xy}}{\sqrt{x}+\sqrt{y}}:\left(\frac{x\sqrt{x}\left(\sqrt{y}-\sqrt{x}\right)}{\sqrt{xy}\left(\sqrt{y}+\sqrt{x}\right)\left(\sqrt{y}-\sqrt{x}\right)}+\frac{y\sqrt{y}\left(\sqrt{x}+\sqrt{y}\right)}{\sqrt{xy}\left(\sqrt{y}+\sqrt{x}\right)\left(\sqrt{y}-\sqrt{x}\right)}-\frac{\left(x+y\right)\left(y-x\right)}{\sqrt{xy}\left(\sqrt{y}+\sqrt{x}\right)\left(\sqrt{y}-\sqrt{x}\right)}\right)\)
\(=\frac{x+y}{\sqrt{x}+\sqrt{y}}:\left(\frac{x\sqrt{xy}-x^2+y\sqrt{xy}+y^2-\left(y^2-x^2\right)}{\sqrt{xy}\left(y-x\right)}\right)\)
\(=\frac{x+y}{\sqrt{x}+\sqrt{y}}:\left(\frac{x\sqrt{xy}+y\sqrt{xy}}{\sqrt{xy}\left(y-x\right)}\right)\)
\(=\frac{x+y}{\sqrt{x}+\sqrt{y}}:\frac{\sqrt{xy}\left(x+y\right)}{\sqrt{xy}\left(y-x\right)}\)
\(=\frac{x+y}{\sqrt{x}+\sqrt{y}}:\frac{x+y}{\left(\sqrt{y}+\sqrt{x}\right)\left(\sqrt{y}-\sqrt{x}\right)}\)
\(=\frac{x+y}{\sqrt{y}+\sqrt{x}}\cdot\frac{\left(\sqrt{y}+\sqrt{x}\right)\left(\sqrt{y}-\sqrt{x}\right)}{x+y}\)
\(=\sqrt{y}-\sqrt{x}\)
Đúng 0
Bình luận (0)
C/m các biể thức sau ko phụ thuộc và giá trị của biến thuộc ĐKXĐ:
a) \(\left(\frac{x\sqrt{x}+y\sqrt{y}}{\sqrt{x}+\sqrt{y}}-\sqrt{xy}\right):\left(x-y\right)+\frac{2\sqrt{y}}{\sqrt{x}+\sqrt{y}}\)
b) \(\left(\frac{\sqrt{x}}{\sqrt{xy-y}}+\frac{2\sqrt{x}+\sqrt{y}}{\sqrt{xy}-x}\right).\frac{x\sqrt{y}-y\sqrt{x}}{x+2\sqrt{xy}+y}\)
Rút gọn:
\(A=\frac{\sqrt{x}-\sqrt{y}}{xy\sqrt{xy}}:\left[\left(\frac{1}{x}+\frac{1}{y}\right).\frac{1}{x+y+2\sqrt{xy}}+\frac{2}{\left(\sqrt{x}+\sqrt{y}\right)^3}.\left(\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{y}}\right)\right]\)
\(x=\sqrt{2-\sqrt{3}};y=\sqrt{2+\sqrt{3}}\)
\(\left(\sqrt{x}+\frac{y-\sqrt{y}}{\sqrt{x}+\sqrt{y}}\right):\left(\right)\frac{x}{\sqrt{xy}+y}+\frac{y}{\sqrt{xy}-x}-\frac{x+y}{\sqrt{xy}}\left(\right)\)
rút gọn tính khi x=3, y=\(4+2\sqrt{3}\)
CẦN GẤP
\(=\dfrac{x+\sqrt{xy}+y-\sqrt{xy}}{\sqrt{x}+\sqrt{y}}:\left(\dfrac{x}{\sqrt{y}\left(\sqrt{x}+\sqrt{y}\right)}-\dfrac{y}{\sqrt{x}\left(\sqrt{x}-\sqrt{y}\right)}-\dfrac{x+y}{\sqrt{xy}}\right)\)
\(=\dfrac{x+y}{\sqrt{x}+\sqrt{y}}:\dfrac{x\sqrt{x}\left(\sqrt{x}-\sqrt{y}\right)-y\sqrt{y}\left(\sqrt{x}+\sqrt{y}\right)-\left(x^2-y^2\right)}{\sqrt{xy}\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}\)
\(=\dfrac{x+y}{\sqrt{x}+\sqrt{y}}\cdot\dfrac{\sqrt{xy}\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}{x^2-x\sqrt{xy}-y\sqrt{xy}-y^2-x^2+y^2}\)
\(=\dfrac{\sqrt{xy}\left(x+y\right)\cdot\left(\sqrt{x}-\sqrt{y}\right)}{-\sqrt{xy}\left(x+y\right)}=-\sqrt{x}+\sqrt{y}\)(1)
Khi x=3 và \(y=4+2\sqrt{3}\) vào (1), ta được:
\(=-\sqrt{3}+\sqrt{4+2\sqrt{3}}=-\sqrt{3}+\sqrt{3}+1=1\)
Đúng 0
Bình luận (0)