Упр.105 ГДЗ Алимов 10-11 класс (Алгебра)

Упростить:

1)

$$\frac{ab^{\frac32}-b^{\frac12}}{a^{\frac12}b^{\frac12}-1} = \frac{b^{\frac12}\left(ab-1\right)}{a^{\frac12}b^{\frac12}-1} = \frac{b^{\frac12}\left(a^{\frac12}b^{\frac12}-1\right)\left(a^{\frac12}b^{\frac12}+1\right)}{a^{\frac12}b^{\frac12}-1} = b^{\frac12}\left(a^{\frac12}b^{\frac12}+1\right) = a^{\frac12}b+b^{\frac12}.$$

Ответ: $$a^{\frac12}b+b^{\frac12}.$$

2)

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

Ответ: $$\frac{\sqrt{ab}}{a-b}.$$