Упр.545 ГДЗ Колягин Ткачёва 9 класс (Алгебра)

1. Сократить дробь: 1) $$\frac{a+\sqrt{3}}{a^2-3}$$; 2) $$\frac{x-\sqrt{2}}{x^2-2}$$; 3) $$\frac{y-9y^{1/2}}{y^{1/4}+3}$$; 4) $$\frac{x+x^{1/2}}{x-1}$$.

1. $$\frac{a+\sqrt{3}}{a^2-3}=\frac{a+\sqrt{3}}{(a-\sqrt{3})(a+\sqrt{3})}=\frac{1}{a-\sqrt{3}}.$$
2. $$\frac{x-\sqrt{2}}{x^2-2}=\frac{x-\sqrt{2}}{(x-\sqrt{2})(x+\sqrt{2})}=\frac{1}{x+\sqrt{2}}.$$
3. $$\frac{y-9y^{1/2}}{y^{1/4}+3}=\frac{y^{1/2}(y^{1/2}-9)}{y^{1/4}+3}.$$
   Представим разность квадратов:
   $$y^{1/2}-9=\left(y^{1/4}-3\right)\left(y^{1/4}+3\right).$$
   Тогда
   $$\frac{y^{1/2}(y^{1/2}-9)}{y^{1/4}+3} = \frac{y^{1/2}(y^{1/4}-3)(y^{1/4}+3)}{y^{1/4}+3} = y^{1/2}(y^{1/4}-3).$$
4. $$\frac{x+x^{1/2}}{x-1}=\frac{x^{1/2}(x^{1/2}+1)}{(x^{1/2}-1)(x^{1/2}+1)}=\frac{x^{1/2}}{x^{1/2}-1}=\frac{\sqrt{x}}{\sqrt{x}-1}.$$

Ответ

1) $$\frac{1}{a-\sqrt{3}}$$;
2) $$\frac{1}{x+\sqrt{2}}$$;
3) $$y^{1/2}(y^{1/4}-3)$$;
4) $$\frac{\sqrt{x}}{\sqrt{x}-1}$$.