Упр.403 ГДЗ Никольский Потапов 9 класс (Алгебра)
- а) $$\left(a^{1/2}+b^{1/2}\right)^{-2}+\left(a^{1/2}-b^{1/2}\right)^{-2}$$
б) $$\left(x^{-1/2}y^{-1/2}+x^{-1/6}\left(y^{-5/6}-x^{-1/3}y^{-1/2}\right)\right)^{-3/2}$$
в) $$\left(\frac{a+1}{a^{2/3}-a^{1/3}+1}-\frac{a-1}{a-a^{1/2}}\right)^2$$
а)
$$\left(a^{-\frac12}+b^{-\frac12}\right)^{-2}+\left(a^{-\frac12}-b^{-\frac12}\right)^{-2}$$
$$=\frac{1}{\left(a^{-\frac12}+b^{-\frac12}\right)^2}+\frac{1}{\left(a^{-\frac12}-b^{-\frac12}\right)^2}$$
$$=\frac{\left(a^{-\frac12}-b^{-\frac12}\right)^2+\left(a^{-\frac12}+b^{-\frac12}\right)^2}{\left(a^{-\frac12}+b^{-\frac12}\right)^2\left(a^{-\frac12}-b^{-\frac12}\right)^2}$$
$$=\frac{2a^{-1}+2b^{-1}}{\left(a^{-1}-b^{-1}\right)^2}$$
$$=\frac{2\left(a^{-1}+b^{-1}\right)}{\left(a^{-1}-b^{-1}\right)^2}.$$б)
$$\left(x^{-\frac12}y^{-\frac12}+x^{-\frac16}\left(y^{-\frac56}-x^{-\frac13}y^{-\frac12}\right)\right)^{-\frac32}$$
$$=\left(x^{-\frac12}y^{-\frac12}+x^{-\frac16}y^{-\frac56}-x^{-\frac12}y^{-\frac12}\right)^{-\frac32}$$
$$=\left(x^{-\frac16}y^{-\frac56}\right)^{-\frac32}$$
$$=x^{\frac14}y^{\frac54}=y\sqrt[4]{xy}.$$в)
$$\left(\frac{a+1}{a^{\frac23}-a^{\frac13}+1}-\frac{a-1}{a-a^{\frac12}}\right)^2$$
$$=\left(\frac{(a^{\frac13}+1)(a^{\frac23}-a^{\frac13}+1)}{a^{\frac23}-a^{\frac13}+1}-\frac{(a^{\frac12}-1)(a^{\frac12}+1)}{a^{\frac12}(a^{\frac12}-1)}\right)^2$$
$$=\left(a^{\frac13}+1-\frac{a^{\frac12}+1}{a^{\frac12}}\right)^2$$
$$=\left(a^{\frac13}-\frac{1}{a^{\frac12}}\right)^2$$
$$=\left(\frac{a^{\frac56}-1}{a^{\frac12}}\right)^2 =\frac{\left(a^{\frac56}-1\right)^2}{a}.$$
Ответ
а) $$\frac{2\left(a^{-1}+b^{-1}\right)}{\left(a^{-1}-b^{-1}\right)^2}$$; б) $$y\sqrt[4]{xy}$$; в) $$\frac{\left(a^{\frac56}-1\right)^2}{a}$$.
