01Математика - 9 класс. Алгебра - Степень с рациональным показателем (нахождение значений выражений, содержащих рациональные степени)

Задание

Найдите значение выражения \(\displaystyle \left(3\frac{3}{8}\right)^{-\frac{1}{3}}\cdot \left(3\frac{13}{81}\right)^{-0{,}{25}}\small.\)

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 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

Сначала найдем \(\displaystyle \left(3\frac{3}{8}\right)^{-\frac{1}{3}}\small,\) затем \(\displaystyle \left(3\frac{13}{81}\right)^{-0{,}{25}} \small, \) потом найдем произведение.

Получаем

\(\displaystyle \left(3\frac{3}{8}\right)^{-\frac{1}{3}}=\left(3\frac{3}{8}\right)^{\frac{\color{blue}{-1}}{\color{red}{3}}}=\sqrt[{\color{red}{3}}]{\left(3\frac{3}{8}\right)^\color{blue}{-1}}=\sqrt[3]{\left(\frac{27}{8}\right)^{-1}}=\sqrt[3]{\frac{8}{27}}=\frac{2}{3}\small,\)

\(\displaystyle \left(3\frac{13}{81}\right)^{-0{,}{25}}=\left(3\frac{13}{81}\right)^{-\frac{1}{4}}=\left(3\frac{13}{81}\right)^{\frac{\color{blue}{-1}}{\color{red}{4}}}=\sqrt[{\color{red}{4}}]{\left(3\frac{13}{81}\right)^\color{blue}{-1}}=\sqrt[{{4}}]{\left(\frac{256}{81}\right)^{-1}}=\sqrt[4]{\frac{81}{256}}=\frac{3}{4}\small,\)

\(\displaystyle \left(3\frac{3}{8}\right)^{-\frac{1}{3}}\cdot \left(3\frac{13}{81}\right)^{-0{,}{25}}=\frac{2}{3}\cdot \frac{3}{4}=\frac{1}{2}=0{,}{5}\small.\)

Ответ: \(\displaystyle 0{,}{5}\small.\)

Теория: 04 Степень с рациональным показателем (нахождение значений выражений, содержащих рациональные степени)

Таблица степеней

С т е п е н и

Ч и с л а

Ч
и
с
л
а