rút gọn biểu thức sau ; \(\frac{x}{\left(\sqrt{x}+\sqrt{y}\right)\left(1-\sqrt{y}\right)}-\frac{y}{\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}+1\right)}-\frac{xy}{\left(\sqrt{x}+1\right)\left(1-\sqrt{y}\right)}\)
\(\frac{X}{\left(\sqrt{X}+\sqrt{Y}\right)\left(1-\sqrt{Y}\right)}-\frac{Y}{\left(\sqrt{X}+\sqrt{Y}\right)\left(\sqrt{X}+1\right)}-\frac{XY}{\left(\sqrt{X}+1\right)\left(1-\sqrt{Y}\right)}\)
Rút gon biểu thức trên
Tìm giá trị nguyên x; y thỏa mãn P=2
\(\text{méo biết}\)
= căn xy + căn x + căn y còn lại tự tính
rút gọn biểu thức:
\(A=\frac{\sqrt{x}-\sqrt{y}}{xy\sqrt{xy}}:\left[\left(\frac{1}{x}+\frac{1}{y}\right).\frac{1}{x+y+2\sqrt{xy}}+\frac{2}{\left(\sqrt{x}+\sqrt{y}\right)^3}.\left(\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{y}}\right)\right]\)
Rút gọn :\(\frac{x}{\left(\sqrt{x}+\sqrt{y}\right).\left(1-\sqrt{y}\right)}-\frac{y}{\left(\sqrt{x}+\sqrt{y}\right).\left(\sqrt{x}+1\right)}-\frac{xy}{\left(\sqrt{x}+1\right).\left(1-\sqrt{y}\right)}\)
Rút gọn biểu thức:
\(C=\frac{\sqrt{x}-\sqrt{y}}{xy\sqrt{xy}}:\left[\left(\frac{1}{x}+\frac{1}{y}\right).\frac{1}{x+y+2\sqrt{xy}}+\frac{2}{\left(\sqrt{x}+\sqrt{y}\right)^3}.\left(\frac{1}{\sqrt{x}+\sqrt{y}}\right)\right]\)
với x=2-\(\sqrt{3}\) và y=2+\(\sqrt{3}\)
P=\(\frac{x}{\left(\sqrt{x}+\sqrt{y}\right)\left(1-\sqrt{y}\right)}-\frac{y}{\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}+1\right)}-\frac{xy}{\left(\sqrt{x}+1\right)\left(1-\sqrt{y}\right)}\)
a.Rút gọn P
b.Tìm x,y nguyên để P=2
Cho biểu thức :
\(P=\frac{x}{\left(\sqrt{x}+\sqrt{y}\right)\left(1-\sqrt{y}\right)}-\frac{y}{\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}+1\right)}-\frac{xy}{\left(\sqrt{x}+1\right)\left(1-\sqrt{y}\right)}\)
a) Tìm ĐKXĐ của x và y để P xác định . Rút gọn P
b) Tìm x , y nguyên thỏa mãn phương trình P = 2
a) ĐKXĐ : \(x,y\ge0;y\ne1;x+y\ne0\)
\(P=\frac{x}{\left(\sqrt{x}+\sqrt{y}\right)\left(1-\sqrt{y}\right)}-\frac{y}{\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}+1\right)}-\frac{xy}{\left(\sqrt{x}+1\right)\left(1-\sqrt{y}\right)}\)
\(=\frac{x\left(1+\sqrt{x}\right)-y\left(1-\sqrt{y}\right)-xy\left(\sqrt{x}+\sqrt{y}\right)}{\left(\sqrt{x}+\sqrt{y}\right)\left(1+\sqrt{x}\right)\left(1-\sqrt{y}\right)}\)
\(=\frac{\left(x-y\right)+\left(x\sqrt{x}+y\sqrt{y}\right)-xy\left(\sqrt{x}+\sqrt{y}\right)}{\left(\sqrt{x}+\sqrt{y}\right)\left(1+\sqrt{x}\right)\left(1-\sqrt{y}\right)}\)
\(=\frac{\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}+x-\sqrt{xy}+y-xy\right)}{\left(\sqrt{x}+\sqrt{y}\right)\left(1+\sqrt{x}\right)\left(1-\sqrt{y}\right)}\)
\(=\frac{\sqrt{x}\left(\sqrt{x}+1\right)-\sqrt{y}\left(\sqrt{x}+1\right)+y\left(1+\sqrt{x}\right)\left(1-\sqrt{x}\right)}{\left(1+\sqrt{x}\right)\left(1+\sqrt{y}\right)}\)
\(=\frac{\sqrt{x}-\sqrt{y}+x-y\sqrt{x}}{1-\sqrt{y}}=\frac{\sqrt{x}\left(1-\sqrt{y}\right)\left(1+\sqrt{y}\right)-\sqrt{y}\left(1-\sqrt{y}\right)}{1-\sqrt{y}}\)
\(=\sqrt{x}+\sqrt{xy}+\sqrt{y}\)
Vậy P \(=\sqrt{x}+\sqrt{xy}+\sqrt{y}\)
b) ĐKXĐ : \(x,y\ge0;y\ne1;x+y\ne0\)
\(P=2\Leftrightarrow\) \(\sqrt{x}+\sqrt{xy}+\sqrt{y}=2\) ( * )
\(\Leftrightarrow\sqrt{x}\left(1+\sqrt{y}\right)-\left(\sqrt{y}+1\right)=1\)
\(\Leftrightarrow\left(\sqrt{x}-1\right)\left(\sqrt{y}+1\right)=1\)
Có : \(1+\sqrt{y}\ge1\Rightarrow\sqrt{x}-1\le1\Leftrightarrow0\le x\le4\Rightarrow x=0;1;2;3;4\)
Thay x = 0 ; 1 ; 2 ; 3 ;4 vào ( * )
Ta có các cặp giá trị : x =4 ; y = 0 và x = 2 ; y = 2 ( TM )
Rút gọn biểu thức: A=\(\frac{\sqrt{x}-\sqrt{y}}{xy\sqrt{xy}}:\left(\left(\frac{1}{x}+\frac{1}{y}\right).\frac{1}{x+y+2\sqrt{xy}}+\frac{2}{\left(\sqrt{x}+\sqrt{y}\right)^3}.\left(\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{y}}\right)\right)\)
Với \(x=2-\sqrt{3}\)
\(y=2+\sqrt{3}\)
Rút gọn biểu thức
a,\(\frac{1}{\left(2\sqrt{x}-2\right)}-\frac{1}{\left(2\sqrt{x}+2\right)}+\frac{\sqrt{x}}{\left(1-x\right)}\)
b, \(\left(\dfrac{\sqrt{x}-\sqrt{y}}{1+\sqrt{xy}}+\dfrac{\sqrt{x}+\sqrt{y}}{1-\sqrt{xy}}\right):\left(\dfrac{x+y+2xy}{1-xy}+1\right)\)
c, \(\dfrac{3\left(x+\sqrt{x}-3\right)}{x+\sqrt{x}-2}+\dfrac{\sqrt{x}+3}{\sqrt{x}+2}-\dfrac{\sqrt{x}-2}{\sqrt{x}-1}\)
a: \(=\dfrac{\sqrt{x}+1-\sqrt{x}+1-2\sqrt{x}}{x-1}=\dfrac{-2\left(\sqrt{x}-1\right)}{x-1}=\dfrac{-2}{\sqrt{x}+1}\)
b: \(=\dfrac{\sqrt{x}-x\sqrt{y}-\sqrt{y}+y\sqrt{x}+\sqrt{x}+x\sqrt{y}+\sqrt{y}+y\sqrt{x}}{1-xy}:\left(\dfrac{x+y+2xy+1-xy}{1-xy}\right)\)
\(=\dfrac{2\sqrt{x}+2y\sqrt{x}}{1-xy}\cdot\dfrac{1-xy}{x+y+xy+1}\)
\(=\dfrac{2\sqrt{x}\left(y+1\right)}{\left(y+1\right)\left(x+1\right)}=\dfrac{2\sqrt{x}}{x+1}\)
c: \(=\dfrac{3x+3\sqrt{x}-9+x+2\sqrt{x}-3-x+4}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}\)
\(=\dfrac{3x+5\sqrt{x}-8}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}=\dfrac{3\sqrt{x}+8}{\sqrt{x}+2}\)
