Question

Cho biểu thức:

N = \(\dfrac{\left(\sqrt{x}-\sqrt{y}\right)^2+\sqrt{xy}}{\sqrt{x}+\sqrt{y}}:\left(\dfrac{x-y}{\sqrt{x}-\sqrt{y}}+\dfrac{x\sqrt{x}-y\sqrt{y}}{y-x}\right)\)

a) Rút gọn N

b) CM: N > 1

Answer 1

a: \(N=\dfrac{x-\sqrt{xy}+y}{\sqrt{x}+\sqrt{y}}:\left(\dfrac{\left(x-y\right)\left(\sqrt{x}+\sqrt{y}\right)-x\sqrt{x}+y\sqrt{y}}{x-y}\right)\)

\(=\dfrac{x-\sqrt{xy}+y}{\sqrt{x}+\sqrt{y}}:\dfrac{x\sqrt{x}+x\sqrt{y}-y\sqrt{x}-y\sqrt{y}-x\sqrt{x}+y\sqrt{y}}{x-y}\)

\(=\dfrac{x-\sqrt{xy}+y}{\sqrt{x}+\sqrt{y}}\cdot\dfrac{x-y}{x\sqrt{y}-y\sqrt{x}}\)

\(=\dfrac{x-\sqrt{xy}+y}{1}\cdot\dfrac{\sqrt{x}-\sqrt{y}}{\sqrt{xy}\left(\sqrt{x}-\sqrt{y}\right)}\)

\(=\dfrac{x-\sqrt{xy}+y}{\sqrt{xy}}\)

b: \(N-1=\dfrac{x-2\sqrt{xy}+y}{\sqrt{xy}}=\dfrac{\left(\sqrt{x}-\sqrt{y}\right)^2}{\sqrt{xy}}>0\)

=>N>1