Задание
Найдите значение выражения \(\displaystyle \sqrt[3]{-27 \cdot 512}{\small .}\)
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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
Заметим, что
\(\displaystyle \sqrt[3\, ] {-27 \cdot 512}=-\sqrt[3\, ] {27 \cdot 512} {\small .}\)
Правило
При \(\displaystyle a < 0{\small }\) и \(\displaystyle k \in \N\)
\(\displaystyle \sqrt[2k+1]{a}=-\sqrt[2k+1]{-a}{\small .}\)
По свойству арифметического корня
\(\displaystyle \sqrt[3 ] {27 \cdot 512}=\sqrt[3] {27} \cdot \sqrt[3 ] {512} {\small .}\)
\(\displaystyle \sqrt[n\,]{ab}=\sqrt[n]{a} \cdot \sqrt[n]{b}{\small}\) при \(\displaystyle a \geqslant 0 {\small,}\)\(\displaystyle b \geqslant 0{\small,}\)\(\displaystyle n \in \N{\small}\)
\(\displaystyle \sqrt[3] {27}=3{\small ,}\,\,\sqrt[3] {512}=8{\small .}\)
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,
\(\displaystyle \sqrt[3\, ] {-27 \cdot 512}=-\sqrt[3\, ] {27 \cdot 512}=-\sqrt[3]{27} \cdot \sqrt[3]{512}=-3 \cdot 8=-24{\small .}\)
Ответ: \(\displaystyle -24{\small .}\)