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\(Q=\frac{x-y}{\sqrt{x}-\sqrt{y}}-\frac{\sqrt{x^3}-\sqrt{y^3}}{x-y}\)
\(Q=\frac{\left(\sqrt{x}+\sqrt{y}\right)\left(x-y\right)-x\sqrt{x}+y\sqrt{y}}{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}\)
\(Q=\frac{x\sqrt{x}-y\sqrt{x}+x\sqrt{y}-y\sqrt{y}-x\sqrt{x}+y\sqrt{y}}{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}\)
\(Q=\frac{\sqrt{xy}\left(\sqrt{x}-\sqrt{y}\right)}{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}\)
\(Q=\frac{\sqrt{xy}}{\sqrt{x}+\sqrt{y}}\)
\(R=\left(\frac{1-a\sqrt{a}}{1-\sqrt{a}}+\sqrt{a}\right).\frac{\left(1-\sqrt{a}\right)^2}{\left(1-a\right)^2}\)
\(R=\left[\frac{\left(1-\sqrt{a}\right)\left(1+\sqrt{a}+a\right)}{1-\sqrt{a}}+\sqrt{a}\right].\frac{\left(1-\sqrt{a}\right)^2}{\left(1-a\right)^2}\)
\(R=\left(1+\sqrt{a}+a\right).\frac{\left(1-\sqrt{a}\right)^2}{\left(1-\sqrt{a}\right)^2.\left(1+\sqrt{a}\right)^2}\)
\(=\left(1+\sqrt{a}\right)^2.\frac{1}{\left(1+\sqrt{a}\right)^2}=1\)
\(\left(\sqrt{x}+\sqrt{y}\right)\left(\frac{x\sqrt{y}-y\sqrt{x}}{\sqrt{xy}}\right)\)
\(=\left(\sqrt{x}+\sqrt{y}\right).\frac{x\sqrt{y}-y\sqrt{x}}{\sqrt{xy}}\)
\(=\frac{-y+\sqrt{x}.\sqrt{y}}{\sqrt{y}}.\left(\sqrt{x}+\sqrt{y}\right)\)
\(=\frac{\left(\sqrt{x}.\sqrt{y}-y\right)\left(\sqrt{x}+\sqrt{y}\right)}{\sqrt{y}}\)
\(=\frac{xy-y^2}{y}\)
\(=\frac{y\left(x-y\right)}{y}\)
= x - y (đpcm)
\(P=\frac{\left(\sqrt{x}+\sqrt{y}\right)\left(1+\sqrt{xy}\right)+\left(\sqrt{x}-\sqrt{y}\right)\left(1-\sqrt{xy}\right)}{\left(1-\sqrt{xy}\right)\left(1+\sqrt{xy}\right)}:\frac{1-xy+x+y+2xy}{\left(1-\sqrt{xy}\right)\left(1+\sqrt{xy}\right)}.\)
\(P=\frac{\sqrt{x}+x\sqrt{y}+\sqrt{y}+y\sqrt{x}+\sqrt{x}-x\sqrt{y}-\sqrt{y}+y\sqrt{x}}{1+x+y+xy}\)
\(P=\frac{2\sqrt{x}}{1+x+y+xy}\)Với ĐK \(x\ge0\) và \(y\ge0\)Và \(xy\ne1\)
Nguyễn Ngọc Anh Minh bạn làm sai rồi kìa bước cuối cùng vẫn còn \(2y\sqrt{x}\)
Ta có : \(P=\frac{\frac{\left(x-y\right)^3}{\left(\sqrt{x}+\sqrt{y}\right)^3}+2x\sqrt{x}+y\sqrt{y}}{x\sqrt{x}+y\sqrt{y}}+\frac{3\left(\sqrt{xy}-y\right)}{x-y}\)
=> \(P=\frac{\frac{\left(\sqrt{x}+\sqrt{y}\right)^3\left(\sqrt{x}-\sqrt{y}\right)^3}{\left(\sqrt{x}+\sqrt{y}\right)^3}+2x\sqrt{x}+y\sqrt{y}}{\sqrt{x}^3+\sqrt{y}^3}+\frac{3\sqrt{y}\left(\sqrt{x}-\sqrt{y}\right)}{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}\)
=> \(P=\frac{\left(\sqrt{x}-\sqrt{y}\right)^3+2x\sqrt{x}+y\sqrt{y}}{\sqrt{x}^3+\sqrt{y}^3}+\frac{3\sqrt{y}}{\sqrt{x}+\sqrt{y}}\)
=> \(P=\frac{x\sqrt{x}-3x\sqrt{y}+3y\sqrt{x}-y\sqrt{y}+2x\sqrt{x}+y\sqrt{y}}{\left(x+y\right)\left(x-\sqrt{xy}+y\right)}+\frac{3\sqrt{y}}{\sqrt{x}+\sqrt{y}}\)
=> \(P=\frac{3x\sqrt{x}-3x\sqrt{y}+3y\sqrt{x}}{\left(x+y\right)\left(x-\sqrt{xy}+y\right)}+\frac{3\sqrt{y}}{\sqrt{x}+\sqrt{y}}\)
=> \(P=\frac{3\sqrt{x}\left(x-\sqrt{xy}+y\right)}{\left(\sqrt{x}+\sqrt{y}\right)\left(x-\sqrt{xy}+y\right)}+\frac{3\sqrt{y}}{\sqrt{x}+\sqrt{y}}\)
=> \(P=\frac{3\sqrt{x}}{\sqrt{x}+\sqrt{y}}+\frac{3\sqrt{y}}{\sqrt{x}+\sqrt{y}}\)
=> \(P=\frac{3\sqrt{x}+3\sqrt{y}}{\sqrt{x}+\sqrt{y}}=\frac{3\left(\sqrt{x}+\sqrt{y}\right)}{\sqrt{x}+\sqrt{y}}=3\)
a) \(\frac{\left(\sqrt{x}\right)^3-\left(\sqrt{y}\right)^3}{\left(\sqrt{x}-\sqrt{y}\right)}-\left(\sqrt{x}-\sqrt{y}\right)^2=\frac{\left(\sqrt{x}-\sqrt{y}\right)\left(x+\sqrt{xy}+y\right)}{\sqrt{x}-\sqrt{y}}-x+2\sqrt{xy}-y\)
\(=3\sqrt{xy}\)
b) \(\frac{x-y}{\sqrt{y}-1}.\sqrt{\frac{\left(\sqrt{y}-1\right)^4}{\left(x-1\right)^4}}=\frac{x-y}{\sqrt{y}-1}.\frac{\left(\sqrt{y}-1\right)^2}{\left(x-1\right)^2}=\frac{\left(x-y\right)\left(\sqrt{y}-1\right)}{\left(x-1\right)^2}\)
a) \(=\frac{\left(\sqrt{x}\right)^3-\left(\sqrt{y}\right)^3}{\sqrt{x}-\sqrt{y}}-\left(\sqrt{x}-\sqrt{y}\right)^2=x+\sqrt{xy}+y-x+2\sqrt{xy}-y=3\sqrt{xy}\)
Rút gọn
\(\frac{\left(x\sqrt{y}+y\sqrt{x}\right)\left(\sqrt{x}-\sqrt{y}\right)}{\sqrt{x}y}=\frac{\sqrt{x}^3\sqrt{y}-xy+yx-\sqrt{x}\sqrt{y}^3}{\sqrt{x}y}=\frac{x}{\sqrt{y}}-y\)