Вычислите: \frac{√{4 \cdot √{32}}}{√2}

Решение:

$$\frac{√{4 \cdot √{32}}}{√2} = \frac{√{4 \cdot √{16 \cdot 2}}}{√2} = \frac{√{4 \cdot 4√{2}}}{√2} = \frac{√{16√{2}}}{√2} = \frac{(16√{2})^{\frac{1}{2}}}{√2} = \frac{16^{\frac{1}{2}} \cdot (√{2})^{\frac{1}{2}}}{√2} = \frac{4 \cdot 2^{\frac{1}{4}}}{2^{\frac{1}{2}}} = 4 \cdot 2^{\frac{1}{4} - \frac{1}{2}} = 4 \cdot 2^{-\frac{1}{4}} = \frac{4}{2^{\frac{1}{4}}} = \frac{4}{\sqrt[4]{2}}$$

Ответ: $$\frac{4}{\sqrt[4]{2}}$$