Упр.1281 ГДЗ Алимов 10-11 класс (Алгебра)

1281. Упростить выражение.

1)

$$\frac{x^{\frac12}}{1+x^{\frac12}}\left(\frac{x^{\frac12}}{1-x^{\frac12}}-\frac{1}{x^{\frac12}-x}\right) = \frac{x^{\frac12}}{1+x^{\frac12}}\left(\frac{x^{\frac12}}{1-x^{\frac12}}-\frac{1}{x^{\frac12}(1-x^{\frac12})}\right)$$
$$= \frac{x^{\frac12}}{1+x^{\frac12}}\cdot \frac{x-1}{x^{\frac12}(1-x^{\frac12})} = \frac{x-1}{(1+x^{\frac12})(1-x^{\frac12})} = \frac{x-1}{1-x} =-1$$

Ответ: $$-1$$

2)

$$\frac{m+2m^{\frac12}+1}{2m^{\frac12}} \left(\frac{2m^{\frac12}}{m^{\frac12}-1}-\frac{4m^{\frac12}}{m-1}\right) = \frac{(m^{\frac12}+1)^2}{2m^{\frac12}} \left(\frac{2m^{\frac12}(m^{\frac12}+1)-4m^{\frac12}}{m-1}\right)$$
$$= \frac{(m^{\frac12}+1)^2}{2m^{\frac12}} \cdot \frac{2m-2m^{\frac12}}{m-1} = \frac{(m^{\frac12}+1)^2}{2m^{\frac12}} \cdot \frac{2m^{\frac12}(m^{\frac12}-1)}{m-1} = m^{\frac12}+1$$

Ответ: $$\sqrt{m}+1$$