Упр.79 ГДЗ Алимов 10-11 класс (Алгебра)

Вычислить:

1. $$\left(2^{\frac{5}{3}}\cdot 3^{-\frac{1}{3}}-3^{\frac{5}{3}}\cdot 2^{-\frac{1}{3}}\right)\sqrt[3]{6};$$
2. $$\left(5^{\frac{1}{4}}\cdot 2^{\frac{3}{4}}-2^{\frac{1}{4}}\cdot 5^{\frac{3}{4}}\right)\sqrt[4]{1000}.$$

1)

$$\left(2^{\frac53}\cdot 3^{-\frac13}-3^{\frac53}\cdot 2^{-\frac13}\right)\sqrt[3]{6}$$
$$=\left(2^{\frac53}\cdot 3^{-\frac13}-3^{\frac53}\cdot 2^{-\frac13}\right)\cdot 2^{\frac13}3^{\frac13}$$
$$=2^{\frac53+\frac13}3^{-\frac13+\frac13}-3^{\frac53+\frac13}2^{-\frac13+\frac13} =2^2-3^2=4-9=-5$$

2)

$$\left(5^{\frac14}\cdot 2^{\frac34}-2^{\frac14}\cdot 5^{\frac34}\right)\sqrt[4]{1000}$$
$$=\left(5^{\frac14}\cdot 2^{\frac34}-2^{\frac14}\cdot 5^{\frac34}\right)\cdot \sqrt[4]{10^3} =\left(5^{\frac14}\cdot 2^{\frac34}-2^{\frac14}\cdot 5^{\frac34}\right)\cdot 10^{\frac34}$$
$$=5^{\frac14+\frac34}2^{\frac34+\frac34}-2^{\frac14+\frac34}5^{\frac34+\frac34} =5^1\cdot 2^{\frac32}-2^1\cdot 5^{\frac32}$$
$$=5\cdot 2\sqrt2-2\cdot 5\sqrt5$$

Ответ: 1) $$-5$$; 2) $$10\sqrt2-10\sqrt5$$.