Упр.17.10 ГДЗ Мерзляк 10 класс Базовый уровень (Алгебра)

1) v7·7^(1/3)·7^(1/6);
2) (3(3^(1/5))^(1/3)·27^(1/5);
3) ((4v2)^(1/3))/(8(2^(1/3)))^(1/4).

1. $$\sqrt{7}\cdot 7^{\frac13}\cdot 7^{\frac16}=7^{\frac12}\cdot 7^{\frac13}\cdot 7^{\frac16}=7^{\frac12+\frac13+\frac16}=7^1=7.$$
2. $$\sqrt[3]{3\sqrt[5]{3}}\cdot \sqrt[5]{27}=\left(3\cdot 3^{\frac15}\right)^{\frac13}\cdot 27^{\frac15}=\left(3^{1+\frac15}\right)^{\frac13}\cdot \left(3^3\right)^{\frac15}$$
   $$=3^{\frac65\cdot \frac13}\cdot 3^{\frac35}=3^{\frac25}\cdot 3^{\frac35}=3^{\frac25+\frac35}=3^1=3.$$
3. $$\frac{\sqrt[3]{4\sqrt2}}{\sqrt[4]{8\sqrt[3]{2}}} =\frac{\sqrt[3]{2^2\cdot 2^{\frac12}}}{\sqrt[4]{2^3\cdot 2^{\frac13}}} =\frac{\sqrt[3]{2^{\frac52}}}{\sqrt[4]{2^{\frac{10}{3}}}} =\frac{2^{\frac56}}{2^{\frac56}}=1.$$

Ответ

1) $$7$$; 2) $$3$$; 3) $$1$$.