6. Сократите дробь: 1) x-9/√x+3; 2) 5+2√5/√5; 3) a-1/a-2√a+1

1) $$\frac{x-9}{\sqrt{x}+3}$$:

$$\frac{x-9}{\sqrt{x}+3} = \frac{(\sqrt{x})^2 - 3^2}{\sqrt{x}+3} = \frac{(\sqrt{x} - 3)(\sqrt{x} + 3)}{\sqrt{x}+3} = \sqrt{x} - 3$$

Ответ: $$\sqrt{x} - 3$$

2) $$\frac{5+2\sqrt{5}}{\sqrt{5}}$$:

$$\frac{5+2\sqrt{5}}{\sqrt{5}} = \frac{5}{\sqrt{5}} + \frac{2\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{5}\cdot\sqrt{5}}{\sqrt{5}} + 2 = \sqrt{5} + 2$$

Ответ: $$\sqrt{5} + 2$$

3) $$\frac{a-1}{a-2\sqrt{a}+1}$$:

$$\frac{a-1}{a-2\sqrt{a}+1} = \frac{(\sqrt{a})^2 - 1^2}{(\sqrt{a} - 1)^2} = \frac{(\sqrt{a}-1)(\sqrt{a}+1)}{(\sqrt{a}-1)^2} = \frac{\sqrt{a}+1}{\sqrt{a}-1}$$

Ответ: $$\frac{\sqrt{a}+1}{\sqrt{a}-1}$$