24. Упростите выражение $$\frac{x+2\sqrt{x}}{\sqrt{x}-2}:(\frac{\sqrt{x}}{\sqrt{x}-2}-\frac{x-12}{x-4}-\frac{4}{x+2\sqrt{x}})$$

Упростим выражение.

$$\frac{x+2\sqrt{x}}{\sqrt{x}-2}:(\frac{\sqrt{x}}{\sqrt{x}-2}-\frac{x-12}{x-4}-\frac{4}{x+2\sqrt{x}}) = \frac{\sqrt{x}(\sqrt{x}+2)}{\sqrt{x}-2}:(\frac{\sqrt{x}}{\sqrt{x}-2}-\frac{x-12}{(\sqrt{x}-2)(\sqrt{x}+2)}-\frac{4}{\sqrt{x}(\sqrt{x}+2)}) = \frac{\sqrt{x}(\sqrt{x}+2)}{\sqrt{x}-2}:(\frac{\sqrt{x}^{2}(\sqrt{x}+2)-(x-12)\sqrt{x}-4(\sqrt{x}-2)}{\sqrt{x}(\sqrt{x}-2)(\sqrt{x}+2)}) = \frac{\sqrt{x}(\sqrt{x}+2)}{\sqrt{x}-2}:(\frac{x(\sqrt{x}+2)-(x-12)\sqrt{x}-4(\sqrt{x}-2)}{\sqrt{x}(\sqrt{x}-2)(\sqrt{x}+2)}) = \frac{\sqrt{x}(\sqrt{x}+2)}{\sqrt{x}-2}:(\frac{x\sqrt{x}+2x-x\sqrt{x}+12\sqrt{x}-4\sqrt{x}+8}{\sqrt{x}(\sqrt{x}-2)(\sqrt{x}+2)}) = \frac{\sqrt{x}(\sqrt{x}+2)}{\sqrt{x}-2}:(\frac{2x+8\sqrt{x}+8}{\sqrt{x}(\sqrt{x}-2)(\sqrt{x}+2)}) = \frac{\sqrt{x}(\sqrt{x}+2)}{\sqrt{x}-2}:(\frac{2(x+4\sqrt{x}+4)}{\sqrt{x}(\sqrt{x}-2)(\sqrt{x}+2)}) = \frac{\sqrt{x}(\sqrt{x}+2)}{\sqrt{x}-2}:(\frac{2(\sqrt{x}+2)^{2}}{\sqrt{x}(\sqrt{x}-2)(\sqrt{x}+2)}) = \frac{\sqrt{x}(\sqrt{x}+2)}{\sqrt{x}-2}:(\frac{2(\sqrt{x}+2)}{\sqrt{x}(\sqrt{x}-2)}) = \frac{\sqrt{x}(\sqrt{x}+2)}{\sqrt{x}-2} \cdot \frac{\sqrt{x}(\sqrt{x}-2)}{2(\sqrt{x}+2)} = \frac{\sqrt{x}\cdot \sqrt{x}}{2} = \frac{x}{2}$$

Ответ: $$\frac{x}{2}$$