4. Найдите значение выражения: a) \(\frac{3⁷ ⁴ (3⁸)²}{3¹¹}\) ; б) \(\frac{(2⁴)⁸}{2¹⁸ ⁴ 2¹⁶}\); в) \(\frac{(5²)⁴ ⁴ 25}{5¹⁰ ⁴ 5}\).

Решение:

Воспользуемся свойствами степеней: \( (a^m)^n = a^{m \cdot n} \) и \( a^m ⁴ a^n = a^{m+n} \), \( \frac{a^m}{a^n} = a^{m-n} \).

1. \( \frac{3^7 ⁴ (3^8)^2}{3^{11}} = \frac{3^7 ⁴ 3^{8 \cdot 2}}{3^{11}} = \frac{3^7 ⁴ 3^{16}}{3^{11}} = \frac{3^{7+16}}{3^{11}} = \frac{3^{23}}{3^{11}} = 3^{23-11} = 3^{12} \)
2. \( \frac{(2^4)^8}{2^{18} ⁴ 2^{16}} = \frac{2^{4 \cdot 8}}{2^{18+16}} = \frac{2^{32}}{2^{34}} = 2^{32-34} = 2^{-2} = \frac{1}{2^2} = \frac{1}{4} \)
3. \( \frac{(5^2)^4 ⁴ 25}{5^{10} ⁴ 5} = \frac{5^{2 \cdot 4} ⁴ 5^2}{5^{10} ⁴ 5^1} = \frac{5^8 ⁴ 5^2}{5^{10} ⁴ 5^1} = \frac{5^{8+2}}{5^{10+1}} = \frac{5^{10}}{5^{11}} = 5^{10-11} = 5^{-1} = \frac{1}{5} \)

Ответ: a) 3¹²; б) \(\frac{1}{4}\); в) \(\frac{1}{5}\).