Question

Chứng minh giá trị biểu thức C không phụ thuộc vào x, y:

C = \(\dfrac{1}{\left(\dfrac{\sqrt{x}+\sqrt{y}}{\sqrt{x+y}}-\dfrac{\sqrt{x+y}}{\sqrt{x}+\sqrt{y}}\right)^2}-\dfrac{x+y}{2\sqrt{x}\sqrt{y}}-\dfrac{\sqrt{\left(x+y\right)^4}}{4xy}\)

Answer 1

\(C=\dfrac{1}{\left(\dfrac{x+2\sqrt{xy}+y-x-y}{\left(\sqrt{x+y}\right)\left(\sqrt{x}+\sqrt{y}\right)}\right)^2}-\dfrac{x+y}{2\sqrt{xy}}-\dfrac{\left(x+y\right)^2}{4xy}\)

\(=\dfrac{\left(x+y\right)\left(\sqrt{x}+\sqrt{y}\right)^2}{4xy}-\dfrac{\left(x+y\right)^2}{4xy}-\dfrac{x+y}{2\sqrt{xy}}\)

\(=\dfrac{\left(x+y\right)\left(x+y+2\sqrt{xy}\right)-\left(x+y\right)^2}{4xy}-\dfrac{x+y}{2\sqrt{xy}}\)

\(=\dfrac{2\sqrt{xy}\left(x+y\right)}{4xy}-\dfrac{x+y}{2\sqrt{xy}}\)

\(=\dfrac{x+y-x-y}{2\sqrt{xy}}=0\)