Задание
Найдите значение выражения \(\displaystyle \sqrt[3\,]{72} \cdot \sqrt[3\,]{3} {\small.}\)
Таблица степеней
Информация
Таблица степеней
С т е п е н и | ||||||||
| \(\displaystyle \bf \color{blue}{2}\) | \(\displaystyle \bf \color{blue}{3}\) | \(\displaystyle \bf \color{blue}{4}\) | \(\displaystyle \bf \color{blue}{5}\) | \(\displaystyle \bf \color{blue}{6}\) | \(\displaystyle \bf \color{blue}{7}\) | \(\displaystyle \bf \color{blue}{8}\) | ||
Ч | \(\displaystyle \bf \color{blue}{2}\) | \(\displaystyle 4\) | \(\displaystyle 8\) | \(\displaystyle 16\) | \(\displaystyle 32\) | \(\displaystyle 64\) | \(\displaystyle 128\) | \(\displaystyle 256\) |
| \(\displaystyle \bf \color{blue}{3}\) | \(\displaystyle 9\) | \(\displaystyle 27\) | \(\displaystyle 81\) | \(\displaystyle 243\) | \(\displaystyle 729\) | \(\displaystyle 2187\) | \(\displaystyle 6561\) | |
| \(\displaystyle \bf \color{blue}{4}\) | \(\displaystyle 16\) | \(\displaystyle 64\) | \(\displaystyle 256\) | \(\displaystyle 1024\) | \(\displaystyle 4096\) | \(\displaystyle 16384\) | \(\displaystyle \ldots\) | |
| \(\displaystyle \bf \color{blue}{5}\) | \(\displaystyle 25\) | \(\displaystyle 125\) | \(\displaystyle 625\) | \(\displaystyle 3125\) | \(\displaystyle 15625\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) | |
| \(\displaystyle \bf \color{blue}{6}\) | \(\displaystyle 36\) | \(\displaystyle 216\) | \(\displaystyle 1296\) | \(\displaystyle 7776\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) | |
| \(\displaystyle \bf \color{blue}{7}\) | \(\displaystyle 49\) | \(\displaystyle 343\) | \(\displaystyle 2401\) | \(\displaystyle 16807\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) | |
| \(\displaystyle \bf \color{blue}{8}\) | \(\displaystyle 64\) | \(\displaystyle 512\) | \(\displaystyle 4096\) | \(\displaystyle 32768\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) | |
| \(\displaystyle \bf \color{blue}{9}\) | \(\displaystyle 81\) | \(\displaystyle 729\) | \(\displaystyle 6561\) | \(\displaystyle 59049\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) | |
| \(\displaystyle \bf \color{blue}{10}\) | \(\displaystyle 100\) | \(\displaystyle 1000\) | \(\displaystyle 10000\) | \(\displaystyle 100000\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) | |
Решение
По свойству арифметического корня
\(\displaystyle \sqrt[n\,]{ab}=\sqrt[n]{a} \cdot \sqrt[n]{b}{\small}\) при \(\displaystyle a \geqslant 0 {\small,}\)\(\displaystyle b \geqslant 0{\small,}\)\(\displaystyle n \in \N{\small}\)
получаем
\(\displaystyle \sqrt[3\,]{72} \cdot \sqrt[3\,]{3}=\sqrt[3\,]{72 \cdot 3}=\sqrt[3\,]{216}{\small .}\)
\(\displaystyle \sqrt[3\, ] {216}=6 {\small .}\)
Информация
Таблица степеней
С т е п е н и | ||||||||
| \(\displaystyle \bf \color{blue}{2}\) | \(\displaystyle \bf \color{blue}{3}\) | \(\displaystyle \bf \color{blue}{4}\) | \(\displaystyle \bf \color{blue}{5}\) | \(\displaystyle \bf \color{blue}{6}\) | \(\displaystyle \bf \color{blue}{7}\) | \(\displaystyle \bf \color{blue}{8}\) | ||
Ч | \(\displaystyle \bf \color{blue}{2}\) | \(\displaystyle 4\) | \(\displaystyle 8\) | \(\displaystyle 16\) | \(\displaystyle 32\) | \(\displaystyle 64\) | \(\displaystyle 128\) | \(\displaystyle 256\) |
| \(\displaystyle \bf \color{blue}{3}\) | \(\displaystyle 9\) | \(\displaystyle 27\) | \(\displaystyle 81\) | \(\displaystyle 243\) | \(\displaystyle 729\) | \(\displaystyle 2187\) | \(\displaystyle 6561\) | |
| \(\displaystyle \bf \color{blue}{4}\) | \(\displaystyle 16\) | \(\displaystyle 64\) | \(\displaystyle 256\) | \(\displaystyle 1024\) | \(\displaystyle 4096\) | \(\displaystyle 16384\) | \(\displaystyle \ldots\) | |
| \(\displaystyle \bf \color{blue}{5}\) | \(\displaystyle 25\) | \(\displaystyle 125\) | \(\displaystyle 625\) | \(\displaystyle 3125\) | \(\displaystyle 15625\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) | |
| \(\displaystyle \bf \color{blue}{6}\) | \(\displaystyle 36\) | \(\displaystyle 216\) | \(\displaystyle 1296\) | \(\displaystyle 7776\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) | |
| \(\displaystyle \bf \color{blue}{7}\) | \(\displaystyle 49\) | \(\displaystyle 343\) | \(\displaystyle 2401\) | \(\displaystyle 16807\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) | |
| \(\displaystyle \bf \color{blue}{8}\) | \(\displaystyle 64\) | \(\displaystyle 512\) | \(\displaystyle 4096\) | \(\displaystyle 32768\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) | |
| \(\displaystyle \bf \color{blue}{9}\) | \(\displaystyle 81\) | \(\displaystyle 729\) | \(\displaystyle 6561\) | \(\displaystyle 59049\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) | |
| \(\displaystyle \bf \color{blue}{10}\) | \(\displaystyle 100\) | \(\displaystyle 1000\) | \(\displaystyle 10000\) | \(\displaystyle 100000\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) | |
Значит,
\(\displaystyle \sqrt[3\,]{72} \cdot \sqrt[3\,]{3}=\sqrt[3\,]{72 \cdot 3}=\sqrt[3\,]{216}=6{\small .}\)
Ответ: \(\displaystyle 6{\small .}\)