Требуется найти корень третьей степени из отрицательного числа \(\displaystyle -0{,}000343{\small .}\)
ОпределениеКорень нечетной \(\displaystyle n\)-й степени из отрицательного числа
Корнем нечетной \(\displaystyle n\)-й степени из отрицательного числа \(\displaystyle a\) называется число, \(\displaystyle n\)-я степень которого равна \(\displaystyle a{\small .}\) Обозначается данный корень так же, как арифметический.
По правилу
ПравилоКорень нечетной степени из отрицательного числа \(\displaystyle a\) связан с арифметическим корнем следующим равенством:
\(\displaystyle \sqrt[2k+1]{a}=\sqrt[2k+1]{-a}=-\!\sqrt[2k+1]{|a|}{\small ,}\)
где \(\displaystyle a < 0{\small .}\)
получаем
\(\displaystyle \sqrt[3]{-0{,}000343}=-\sqrt[3]{0{,}000343}{\small .}\)
Найдём арифметический корень \(\displaystyle \sqrt[\color{red}{\bf3}]{0{,}000343}{\small ,}\)то есть неотрицательное число, третья степень которого равна \(\displaystyle 0{,}000343{\small .}\)
\(\displaystyle \left(\color{green}{0{,}07}\right)^{\color{red}{3}}= 0{,}000343{\small .}\)
ИнформацияТаблица степеней
| | С т е п е н и |
| \(\displaystyle \bf \color{blue}{2}\) | \(\displaystyle \bf \color{red}{3}\) | \(\displaystyle \bf \color{blue}{4}\) | \(\displaystyle \bf \color{blue}{5}\) | \(\displaystyle \bf \color{blue}{6}\) | \(\displaystyle \bf \color{blue}{7}\) | \(\displaystyle \bf \color{blue}{8}\) |
Ч
и
с
л
а | \(\displaystyle \bf \color{blue}{2}\) | \(\displaystyle 4\) | \(\displaystyle 8\) | \(\displaystyle 16\) | \(\displaystyle 32\) | \(\displaystyle 64\) | \(\displaystyle 128\) | \(\displaystyle 256\) |
| \(\displaystyle \bf \color{blue}{3}\) | \(\displaystyle 9\) | \(\displaystyle 27\) | \(\displaystyle 81\) | \(\displaystyle 243\) | \(\displaystyle 729\) | \(\displaystyle 2187\) | \(\displaystyle 6561\) |
| \(\displaystyle \bf \color{blue}{4}\) | \(\displaystyle 16\) | \(\displaystyle 64\) | \(\displaystyle 256\) | \(\displaystyle 1024\) | \(\displaystyle 4096\) | \(\displaystyle 16384\) | \(\displaystyle \ldots\) |
| \(\displaystyle \bf \color{blue}{5}\) | \(\displaystyle 25\) | \(\displaystyle 125\) | \(\displaystyle 625\) | \(\displaystyle 3125\) | \(\displaystyle 15625\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) |
| \(\displaystyle \bf \color{blue}{6}\) | \(\displaystyle 36\) | \(\displaystyle 216\) | \(\displaystyle 1296\) | \(\displaystyle 7776\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) |
| \(\displaystyle \bf \color{green}{7}\) | \(\displaystyle 49\) | \(\displaystyle \bf{\orange{343}}\) | \(\displaystyle 2401\) | \(\displaystyle 16807\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) |
| \(\displaystyle \bf \color{blue}{8}\) | \(\displaystyle 64\) | \(\displaystyle 512\) | \(\displaystyle 4096\) | \(\displaystyle 32768\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) |
| \(\displaystyle \bf \color{blue}{9}\) | \(\displaystyle 81\) | \(\displaystyle 729\) | \(\displaystyle 6561\) | \(\displaystyle 59049\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) |
| \(\displaystyle \bf \color{blue}{10}\) | \(\displaystyle 100\) | \(\displaystyle 1000\) | \(\displaystyle 10000\) | \(\displaystyle 100000\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) | \(\displaystyle \ldots\) |
Видим, что
\(\displaystyle 7^{\,3}=343{\small .}\)
Значит,
\(\displaystyle \left(0{,}07\right)^{3}= 0{,}000343{\small .}\)
Значит,
\(\displaystyle \sqrt[\color{red}{\bf3}]{0{,}000343}=\color{green}{0{,}07}{\small .}\)
Получаем
\(\displaystyle \sqrt[3]{-0{,}000343}=-\sqrt[3]{0{,}000343}=-0{,}07{\small .}\)
Ответ: \(\displaystyle \sqrt[3]{0{,}000343}=-0{,}07{\small .}\)