Упр.70 ГДЗ Колягин Ткачёва 9 класс (Алгебра)

1. Упростить выражение:
   1) $$a^{1/9}\left(a\left(a^{1/3}\right)\right)^{1/6}$$;
   2) $$\left((ab^{-2})^{1/3}+(ab)^{-1/6}\right)(ab^4)^{1/6}$$;
   3) $$b^{1/12}\left(b\left(b^{1/4}\right)\right)^{1/3}$$;
   4) $$\left(a^{1/3}+b^{1/3}\right)\left(a^{2/3}+b^{2/3}-(ab)^{1/3}\right)$$.

1. $$a^{\frac19}\left(a\left(a^{\frac13}\right)\right)^{\frac16} =a^{\frac19}\left(a^{1+\frac13}\right)^{\frac16} =a^{\frac19}\left(a^{\frac43}\right)^{\frac16} =a^{\frac19}a^{\frac{4}{18}} =a^{\frac{2}{18}+\frac{4}{18}} =a^{\frac13} =\sqrt[3]{a}.$$

2. $$\left(\left(ab^{-2}\right)^{\frac13}+(ab)^{-\frac16}\right)(ab^4)^{\frac16}$$
   $$=\left(a^{\frac13}b^{-\frac23}+a^{-\frac16}b^{-\frac16}\right)a^{\frac16}b^{\frac46}$$
   $$=a^{\frac13+\frac16}b^{-\frac23+\frac46}+a^{-\frac16+\frac16}b^{-\frac16+\frac46}$$
   $$=a^{\frac12}+b^{\frac12} =\sqrt{a}+\sqrt{b}.$$

3. $$b^{\frac{1}{12}}\left(b\left(b^{\frac14}\right)\right)^{\frac13} =b^{\frac{1}{12}}\left(b^{1+\frac14}\right)^{\frac13} =b^{\frac{1}{12}}\left(b^{\frac54}\right)^{\frac13} =b^{\frac{1}{12}}b^{\frac{5}{12}} =b^{\frac{6}{12}} =b^{\frac12} =\sqrt{b}.$$

4. $$\left(\sqrt[3]{a}+\sqrt[3]{b}\right)\left(a^{\frac23}+b^{\frac23}-\sqrt[3]{ab}\right)$$
   $$=\left(\sqrt[3]{a}+\sqrt[3]{b}\right)\left(\left(\sqrt[3]{a}\right)^2+\left(\sqrt[3]{b}\right)^2-\sqrt[3]{a}\sqrt[3]{b}\right)$$
   $$=\left(\sqrt[3]{a}\right)^3+\left(\sqrt[3]{b}\right)^3 =a+b.$$

Ответ

1) $$\sqrt[3]{a}$$; 2) $$\sqrt{a}+\sqrt{b}$$; 3) $$\sqrt{b}$$; 4) $$a+b$$.