Question

Rút gọn: \(\left(\frac{x-y}{\sqrt{x}-\sqrt{y}}-\frac{x\sqrt{x}-y\sqrt{y}}{x-y}\right).\frac{\left(\sqrt{x}-\sqrt{y}\right)^2}{x\sqrt{x}+y\sqrt{y}}\)

Answer 1

Lời giải:

ĐK: \(x,y\geq 0; x\neq y\). Để cho gọn đặt \(\sqrt{x}=a; \sqrt{y}=b\). Khi đó:

\(\left(\frac{x-y}{\sqrt{x}-\sqrt{y}}-\frac{x\sqrt{x}-y\sqrt{y}}{x-y}\right).\frac{(\sqrt{x}-\sqrt{y})^2}{x\sqrt{x}+y\sqrt{y}}\)

\(=(\frac{a^2-b^2}{a-b}-\frac{a^3-b^3}{a^2-b^2}).\frac{(a-b)^2}{a^3+b^3}\)

\(=\frac{(a^2-b^2)(a+b)-(a^3-b^3)}{a^2-b^2}.\frac{(a-b)^2}{a^3+b^3}\)

\(=\frac{ab(a-b)}{(a-b)(a+b)}.\frac{(a-b)^2}{a^3+b^3}=\frac{ab(a-b)^2}{(a+b)(a^3+b^3)}\)

\(=\frac{\sqrt{xy}(\sqrt{x}-\sqrt{y})^2}{(\sqrt{x}+\sqrt{y})(x\sqrt{x}+y\sqrt{y})}\)