` a) ` ` \frac{x - 4}{\sqrt{x} - 2} `
` = \frac{(\sqrt{x} - 2)(\sqrt{x} + 2)}{\sqrt{x} - 2} `
` = \sqrt{x} + 2 `
` b) ` ` \frac{\sqrt{x} + 1}{x - 1} `
` = \frac{\sqrt{x} + 1}{(\sqrt{x} - 1)(\sqrt{x} + 1)} `
` = \frac{1}{\sqrt{x} - 1} `
` c) ` ` \frac{1 - x\sqrt{x}}{1 - \sqrt{x}} `
` = \frac{1^3 - (\sqrt{x})^3}{1 - \sqrt{x}} `
` = \frac{(1 - \sqrt{x})(1 + \sqrt{x} + x)}{1 - \sqrt{x}} `
` = x + \sqrt{x} + 1 `
` d) ` ` \frac{x\sqrt{x} - y\sqrt{y}}{x + \sqrt{xy} + y} `
` = \frac{(\sqrt{x})^3 - (\sqrt{y})^3}{x + \sqrt{xy} + y} `
` = \frac{(\sqrt{x} - \sqrt{y})(x + \sqrt{xy} + y)}{x + \sqrt{xy} + y} `
` = \sqrt{x} - \sqrt{y} `
