Đáp án:
Giải thích các bước giải:
p) Với $x ≥ 0, x \neq 1, x \neq 9$ thì:
P = $(\frac{x + 2\sqrt{x} - 7}{x - 9} + \frac{\sqrt{x} - 1}{3 - \sqrt{x}}) ÷ (\frac{1}{\sqrt{x} + 3} - \frac{1}{\sqrt{x} - 1})$
= $(\frac{x + 2\sqrt{x} - 7}{(\sqrt{x} + 3)(\sqrt{x} - 3)} + \frac{(1 - \sqrt{x})(\sqrt{x} + 3)}{(3 + \sqrt{x})(\sqrt{x} - 3)}) ÷ (\frac{1 - \sqrt{x}}{(\sqrt{x} + 3)(1 - \sqrt{x})} + \frac{\sqrt{x} + 3}{(1 - \sqrt{x})(\sqrt{x} + 3)})$
= $\frac{x + 2\sqrt{x} - 7 + (1 - \sqrt{x})(\sqrt{x} + 3)}{(\sqrt{x} + 3)(\sqrt{x} - 3)}÷\frac{1 - \sqrt{x} + \sqrt{x} + 3}{(\sqrt{x} + 3)(1 - \sqrt{x})}$
= $\frac{x + 2\sqrt{x} - 7 + \sqrt{x} - x - 3\sqrt{x} + 3}{(\sqrt{x} + 3)(\sqrt{x} - 3)}÷\frac{4}{(\sqrt{x} + 3)(1 - \sqrt{x})}$
= $\frac{-4}{(\sqrt{x} + 3)(\sqrt{x} - 3)}÷\frac{4}{(\sqrt{x} + 3)(1 - \sqrt{x})}$
= $\frac{-4}{(\sqrt{x} + 3)(\sqrt{x} - 3)} × \frac{(\sqrt{x} + 3)(1 - \sqrt{x})}{4}$
= $\frac{\sqrt{x} - 1}{(\sqrt{x} - 3)}$
q) Với x > 0, x $\neq$ 1 thì:
Q = $(\frac{x}{\sqrt{x}(\sqrt{x} - 1)} - \frac{1}{\sqrt{x}(\sqrt{x} - 1)})÷(\frac{\sqrt{x} - 1}{(\sqrt{x} - 1)(\sqrt{x} + 1)} + \frac{2}{(\sqrt{x} - 1)(\sqrt{x} + 1)})$
= $\frac{x - 1}{\sqrt{x}(\sqrt{x} - 1)} ÷ \frac{\sqrt{x} + 1}{(\sqrt{x} - 1)(\sqrt{x} + 1)}$
= $\frac{\sqrt{x} + 1}{\sqrt{x}} ÷ \frac{1}{\sqrt{x} - 1}$
= $\frac{x - 1}{\sqrt{x}}$
r) Với x > 0 và x $\neq$ 4 thì:
R = $\frac{\sqrt{x}(\sqrt{x}^{3} - 1)}{x + \sqrt{x} + 1} - \frac{\sqrt{x}(2\sqrt{x}+ 1)}{\sqrt{x}} + \frac{2(\sqrt{x} + 1)(\sqrt{x} - 1)}{\sqrt{x} - 1}$
= $\frac{\sqrt{x}(\sqrt{x} - 1)(x + \sqrt{x} + 1)}{x + \sqrt{x} + 1} - (2\sqrt{x} + 1) + 2(\sqrt{x} + 1)$
= $\sqrt{x}(\sqrt{x} - 1) - 2\sqrt{x} - 1 + 2\sqrt{x} + 2$
= $x - \sqrt{x} - 2\sqrt{x} - 1 + 2\sqrt{x} + 2$
= $x - \sqrt{x} + 1$
s) Với x > 0, x $\neq$ 4 thì:
Q = $(\frac{4\sqrt{x}(\sqrt{x} - 2)}{(2 + \sqrt{x})(2 - \sqrt{x})}+\frac{8x}{(2 + \sqrt{x})(2 - \sqrt{x})})÷(\frac{\sqrt{x} - 1}{\sqrt{x}(\sqrt{x} - 2)} - \frac{2(\sqrt{x} - 2)}{\sqrt{x}(\sqrt{x} - 2)})$
= $\frac{4x - 8\sqrt{x} + 8x}{(2 + \sqrt{x})(2 - \sqrt{x})} ÷ \frac{\sqrt{x} - 1 - 2\sqrt{x} + 4}{\sqrt{x}(\sqrt{x} - 2)}$
= $\frac{12x - 8\sqrt{x}}{(2 + \sqrt{x})(2 - \sqrt{x})} ÷ \frac{- \sqrt{x} + 3}{\sqrt{x}(\sqrt{x} - 2)}$
= $\frac{(12 - 8\sqrt{x})(\sqrt{x} - 3)}{2 + \sqrt{x}}$
= $\frac{12\sqrt{x} - 8x - 36 + 24\sqrt{x}}{2 + \sqrt{x}}$
= $\frac{-8x + 36\sqrt{x} - 36}{2 + \sqrt{x}}$