Найдите показатели степеней:
| \(\displaystyle 4^{12}\cdot 9^{18}\cdot 4^6\cdot 9^7\) | \(\displaystyle =\,\) | \(\displaystyle 4\) |
| \(\displaystyle \cdot\) | \(\displaystyle 9\) |
|
Произведение степеней
Пусть \(\displaystyle a\) – число, \(\displaystyle n,\, m\) – натуральные числа, тогда
\(\displaystyle {\bf a^n\cdot a^m= a^{n+m}}.\)
Менее формально, при умножении степеней с одинаковыми основаниями показатели степеней складываются.
Сначала сгруппируем выражения с одинаковыми основаниями:
\(\displaystyle 4^{12}\cdot 9^{18}\cdot 4^{6}\cdot 9^{7}={\color{blue}4}^{12}\cdot {\color{red}9}^{18}\cdot {\color{blue}4}^{6}\cdot {\color{red}9}^{7}=({\color{blue}4}^{12}\cdot {\color{blue}4}^{6})\cdot ({\color{red}9}^{18}\cdot {\color{red}9}^{7})\),
затем воспользуемся правилом сложения степеней:
\(\displaystyle ({\color{blue}4}^{12}\cdot {\color{blue}4}^{6})\cdot ({\color{red}9}^{18}\cdot {\color{red}9}^{7})={\color{blue}4}^{12+6}\cdot {\color{red}9}^{18+7}={\color{blue}4}^{\bf 18}\cdot {\color{red}9}^{\bf 25}\).
| \(\displaystyle 4^{12}\cdot 9^{18}\cdot 4^{6}\cdot 9^{7}\) | \(\displaystyle =\) | \(\displaystyle \underline{4\cdot 4\cdot \dots \cdot 4}\) | \(\displaystyle \cdot\) | \(\displaystyle \underline{9\cdot 9\cdot \dots \cdot 9}\) | \(\displaystyle \cdot\) | \(\displaystyle \underline{4\cdot 4\cdot \dots \cdot 4}\) | \(\displaystyle \cdot\) | \(\displaystyle \underline{9\cdot 9\cdot \dots \cdot 9}\) | \(\displaystyle =\) |
| \(\displaystyle {\color{blue}{12}}\) раз | \(\displaystyle {\color{red}{18}}\) раз | \(\displaystyle {\color{blue}6}\) раз | \(\displaystyle {\color{red}7}\) раз | → | |||||
| \(\displaystyle =\) | \(\displaystyle (\underline{4\cdot 4\cdot \dots \cdot 4}\) | \(\displaystyle \cdot\) | \(\displaystyle \underline{4\cdot 4\cdot \dots \cdot 4})\) | \(\displaystyle \cdot\) | \(\displaystyle (\underline{9\cdot 9\cdot \dots \cdot 9}\) | \(\displaystyle \cdot\) | \(\displaystyle \underline{9\cdot 9\cdot \dots \cdot 9})\) | \(\displaystyle =\) | |
| \(\displaystyle {\color{blue}{12}}\) раз | \(\displaystyle {\color{blue}6}\) раз | \(\displaystyle {\color{red}{18}}\) раз | \(\displaystyle {\color{red}7}\) раз | → | |||||
| \(\displaystyle =\) | \(\displaystyle \underline{4\cdot 4\cdot \dots \cdot 4}\) | \(\displaystyle \cdot\) | \(\displaystyle \underline{9\cdot 9\cdot \dots \cdot 9}\) | \(\displaystyle =4^{\color{blue}{18}}\cdot 9^{\color{red}{25}}\) | |||||
| \(\displaystyle {\color{blue}{12+6}}\) раз | \(\displaystyle {\color{red}{18+7}}\) раз | ||||||||