Упр.66 ГДЗ Алимов 10-11 класс (Алгебра)

Сократить дробь:

1. а) $$\frac{\sqrt{a}-\sqrt{b}}{a^{1/4}-1/4}$$;
2. 2) $$\frac{\sqrt{m}+\sqrt{n}}{b+2\sqrt{mn}+n}$$;
3. 3) $$\frac{c-2\sqrt{c}+1}{\sqrt{c}-1}$$.

1) $$\frac{\sqrt{a}-\sqrt{b}}{a^{\frac14}-b^{\frac14}}=\frac{a^{\frac12}-b^{\frac12}}{a^{\frac14}-b^{\frac14}}=\frac{\left(a^{\frac14}-b^{\frac14}\right)\left(a^{\frac14}+b^{\frac14}\right)}{a^{\frac14}-b^{\frac14}}=a^{\frac14}+b^{\frac14}.$$

2) $$\frac{m^{\frac12}+n^{\frac12}}{m+2\sqrt{mn}+n}=\frac{m^{\frac12}+n^{\frac12}}{\left(\sqrt{m}+\sqrt{n}\right)^2}=\frac{m^{\frac12}+n^{\frac12}}{\left(m^{\frac14}+n^{\frac14}\right)^2}=\frac{1}{m^{\frac12}+n^{\frac12}}.$$

3) $$\frac{c-2c^{\frac12}+1}{\sqrt{c}-1}=\frac{\left(c^{\frac12}-1\right)^2}{c^{\frac12}-1}=c^{\frac12}-1.$$

Ответ: 1) $$a^{\frac14}+b^{\frac14}$$; 2) $$\frac{1}{m^{\frac12}+n^{\frac12}}$$; 3) $$c^{\frac12}-1$$.