3. Используя свойства степеней, найдите значение вы- ражения: 1) a) 3⁷ · (3²)³ : 3¹⁰; б) 5²⁰ : (5²)⁵ · 5⁸; 2) a) 9⁴/3⁷; б) 8⁵/4⁶; 3) a) 10¹²/2⁶·5⁶; б) 5¹⁶·3¹⁶/15¹⁴; в) 27²·9⁴/81²; 3⁵·4⁵/12⁶

Ответ:

1) a) \(3^7 \cdot (3^2)^3 : 3^{10} = 3^7 \cdot 3^6 : 3^{10} = 3^{7+6-10} = 3^3 = 27\) б) \(5^{20} : (5^2)^5 \cdot 5^8 = 5^{20} : 5^{10} \cdot 5^8 = 5^{20-10+8} = 5^{18} \) 2) a) \(\frac{9^4}{3^7} = \frac{(3^2)^4}{3^7} = \frac{3^8}{3^7} = 3^{8-7} = 3^1 = 3\) б) \(\frac{8^5}{4^6} = \frac{(2^3)^5}{(2^2)^6} = \frac{2^{15}}{2^{12}} = 2^{15-12} = 2^3 = 8\) 3) a) \(\frac{10^{12}}{2^6 \cdot 5^6} = \frac{(2 \cdot 5)^{12}}{2^6 \cdot 5^6} = \frac{2^{12} \cdot 5^{12}}{2^6 \cdot 5^6} = 2^{12-6} \cdot 5^{12-6} = 2^6 \cdot 5^6 = (2 \cdot 5)^6 = 10^6 = 1000000\) б) \(\frac{5^{16} \cdot 3^{16}}{15^{14}} = \frac{(5 \cdot 3)^{16}}{15^{14}} = \frac{15^{16}}{15^{14}} = 15^{16-14} = 15^2 = 225\) в) \(\frac{27^2 \cdot 9^4}{81^2} = \frac{(3^3)^2 \cdot (3^2)^4}{(3^4)^2} = \frac{3^6 \cdot 3^8}{3^8} = 3^6 = 729\) г) \(\frac{3^5 \cdot 4^5}{12^6} = \frac{3^5 \cdot (2^2)^5}{(3 \cdot 4)^6} = \frac{3^5 \cdot 2^{10}}{3^6 \cdot 4^6} = \frac{3^5 \cdot 2^{10}}{3^6 \cdot (2^2)^6} = \frac{3^5 \cdot 2^{10}}{3^6 \cdot 2^{12}} = \frac{1}{3 \cdot 2^2} = \frac{1}{3 \cdot 4} = \frac{1}{12}\)