01Математика - 9 класс. Алгебра - Корень \(n\)-й степени. Вычисление корней нечетных степеней из отрицательных чисел

Задание

Найдите значение выражения

\(\displaystyle \sqrt[5]{-\frac{3025}{7776}}=\)

-\frac{5}{6}

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 и с л а	\(\displaystyle \bf \color{blue}{2}\)	\(\displaystyle 4\)	\(\displaystyle 8\)	\(\displaystyle 16\)	\(\displaystyle 32\)	\(\displaystyle 64\)	\(\displaystyle 128\)	\(\displaystyle 256\)
	\(\displaystyle \bf \color{blue}{3}\)	\(\displaystyle 9\)	\(\displaystyle 27\)	\(\displaystyle 81\)	\(\displaystyle 243\)	\(\displaystyle 729\)	\(\displaystyle 2187\)	\(\displaystyle 6561\)
	\(\displaystyle \bf \color{blue}{4}\)	\(\displaystyle 16\)	\(\displaystyle 64\)	\(\displaystyle 256\)	\(\displaystyle 1024\)	\(\displaystyle 4096\)	\(\displaystyle 16384\)	\(\displaystyle \ldots\)
	\(\displaystyle \bf \color{blue}{5}\)	\(\displaystyle 25\)	\(\displaystyle 125\)	\(\displaystyle 625\)	\(\displaystyle 3125\)	\(\displaystyle 15625\)	\(\displaystyle \ldots\)	\(\displaystyle \ldots\)
	\(\displaystyle \bf \color{blue}{6}\)	\(\displaystyle 36\)	\(\displaystyle 216\)	\(\displaystyle 1296\)	\(\displaystyle 7776\)	\(\displaystyle \ldots\)	\(\displaystyle \ldots\)	\(\displaystyle \ldots\)
	\(\displaystyle \bf \color{blue}{7}\)	\(\displaystyle 49\)	\(\displaystyle 343\)	\(\displaystyle 2401\)	\(\displaystyle 16807\)	\(\displaystyle \ldots\)	\(\displaystyle \ldots\)	\(\displaystyle \ldots\)
	\(\displaystyle \bf \color{blue}{8}\)	\(\displaystyle 64\)	\(\displaystyle 512\)	\(\displaystyle 4096\)	\(\displaystyle 32768\)	\(\displaystyle \ldots\)	\(\displaystyle \ldots\)	\(\displaystyle \ldots\)
	\(\displaystyle \bf \color{blue}{9}\)	\(\displaystyle 81\)	\(\displaystyle 729\)	\(\displaystyle 6561\)	\(\displaystyle 59049\)	\(\displaystyle \ldots\)	\(\displaystyle \ldots\)	\(\displaystyle \ldots\)
	\(\displaystyle \bf \color{blue}{10}\)	\(\displaystyle 100\)	\(\displaystyle 1000\)	\(\displaystyle 10000\)	\(\displaystyle 100000\)	\(\displaystyle \ldots\)	\(\displaystyle \ldots\)	\(\displaystyle \ldots\)

Решение

Требуется найти корень пятой степени из отрицательного числа \(\displaystyle -\frac{3025}{7776}{\small .}\)

Определение

Корень 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

Правило

Корень 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

\(\displaystyle \sqrt[5]{-\frac{3025}{7776}}=-\sqrt[5]{\frac{3025}{7776}}{\small .}\)

Найдём арифметический корень \(\displaystyle \sqrt[\color{red}{\bf5}]{\frac{3025}{7776}}{\small ,}\)то есть неотрицательное число, пятая степень которого равна \(\displaystyle \frac{3025}{7776}{\small .}\)

\(\displaystyle \left(\color{green}{\frac{5}{6}}\right)^{\!\color{red}{5}}=\frac{3025}{7776}{\small .}\)

Воспользуемся таблицей степеней:

Информация

Таблица степеней

		С т е п е н и
		\(\displaystyle \bf \color{blue}{2}\)	\(\displaystyle \bf \color{blue}{3}\)	\(\displaystyle \bf \color{blue}{4}\)	\(\displaystyle \bf \color{red}{5}\)	\(\displaystyle \bf \color{blue}{6}\)	\(\displaystyle \bf \color{blue}{7}\)	\(\displaystyle \bf \color{blue}{8}\)
Ч и с л а	\(\displaystyle \bf \color{blue}{2}\)	\(\displaystyle 4\)	\(\displaystyle 8\)	\(\displaystyle 16\)	\(\displaystyle 32\)	\(\displaystyle 64\)	\(\displaystyle 128\)	\(\displaystyle 256\)
	\(\displaystyle \bf \color{blue}{3}\)	\(\displaystyle 9\)	\(\displaystyle 27\)	\(\displaystyle 81\)	\(\displaystyle 243\)	\(\displaystyle 729\)	\(\displaystyle 2187\)	\(\displaystyle 6561\)
	\(\displaystyle \bf \color{blue}{4}\)	\(\displaystyle 16\)	\(\displaystyle 64\)	\(\displaystyle 256\)	\(\displaystyle 1024\)	\(\displaystyle 4096\)	\(\displaystyle 16384\)	\(\displaystyle \ldots\)
	\(\displaystyle \bf \color{green}{5}\)	\(\displaystyle 25\)	\(\displaystyle 125\)	\(\displaystyle 625\)	\(\displaystyle \bf{\orange{3125}}\)	\(\displaystyle 15625\)	\(\displaystyle \ldots\)	\(\displaystyle \ldots\)
	\(\displaystyle \bf \color{green}{6}\)	\(\displaystyle 36\)	\(\displaystyle 216\)	\(\displaystyle 1296\)	\(\displaystyle \bf{\orange{7776}}\)	\(\displaystyle \ldots\)	\(\displaystyle \ldots\)	\(\displaystyle \ldots\)
	\(\displaystyle \bf \color{blue}{7}\)	\(\displaystyle 49\)	\(\displaystyle 343\)	\(\displaystyle 2401\)	\(\displaystyle 16807\)	\(\displaystyle \ldots\)	\(\displaystyle \ldots\)	\(\displaystyle \ldots\)
	\(\displaystyle \bf \color{blue}{8}\)	\(\displaystyle 64\)	\(\displaystyle 512\)	\(\displaystyle 4096\)	\(\displaystyle 32768\)	\(\displaystyle \ldots\)	\(\displaystyle \ldots\)	\(\displaystyle \ldots\)
	\(\displaystyle \bf \color{blue}{9}\)	\(\displaystyle 81\)	\(\displaystyle 729\)	\(\displaystyle 6561\)	\(\displaystyle 59049\)	\(\displaystyle \ldots\)	\(\displaystyle \ldots\)	\(\displaystyle \ldots\)
	\(\displaystyle \bf \color{blue}{10}\)	\(\displaystyle 100\)	\(\displaystyle 1000\)	\(\displaystyle 10000\)	\(\displaystyle 100000\)	\(\displaystyle \ldots\)	\(\displaystyle \ldots\)	\(\displaystyle \ldots\)

Видим, что

\(\displaystyle 5^{\,5}=3025\) и \(\displaystyle 6^{\,5}=7776{\small .}\)

Правило

Частное в степени

Для любых чисел \(\displaystyle a,\, b=\not 0\)и натурального числа \(\displaystyle n\) верно

\(\displaystyle \frac{a^{\,n}}{b^{\,n}}=\left(\frac{a}{b}\,\right)^n.\)

Значит,

\(\displaystyle \frac{3025}{7776}=\frac{5^{\,5}}{6^{\,5}}=\left(\frac{5}{6}\right)^{\!5} {\small .}\)

Значит,

\(\displaystyle \sqrt[\color{red}{\bf5}]{\frac{3025}{7776}}=\color{green}{\frac{5}{6}}{\small .}\)

Получаем

\(\displaystyle \sqrt[5]{-\frac{3025}{7776}}=-\sqrt[5]{\frac{3025}{7776}}=-\frac{5}{6}{\small .}\)

Ответ: \(\displaystyle \sqrt[4]{\frac{3025}{7776}}=\frac{5}{6}{\small .}\)

Теория: Вычисление корней нечетных степеней из отрицательных чисел

Таблица степеней

С т е п е н и

Ч и с л а

Корень нечетной \(\displaystyle n\)-й степени из отрицательного числа

Таблица степеней

С т е п е н и

Ч и с л а

Частное в степени

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