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

Задание

Найдите значение выражения

\(\displaystyle \left(\left(\frac{1}{16}\right)^{-0{,}{75}}\right)^{\frac{2}{3}}\small.\)

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 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 \(\displaystyle \left(\frac{1}{16}\right)^{-0{,}{75}}\small.\)

\(\displaystyle \left(\frac{1}{16}\right)^{-0{,}{75}}=\left(\frac{1}{16}\right)^{-\frac{3}{4}}=\left(\frac{1}{16}\right)^{\frac{\color{blue}{-3}}{\color{red}{4}}}=\sqrt[{\color{red}{4}}]{\left(\frac{1}{16}\right)^\color{blue}{-3}}=\left(\sqrt[4]{\frac{1}{16}}\right)^{-3}=\left({\frac{1}{2}}\right)^{-3}={8}\small.\)

Тогда

\(\displaystyle \left(\left(\frac{1}{16}\right)^{-0{,}{75}}\right)^{\frac{2}{3}}=8^{\frac{\color{blue}{2}}{\color{red}{3}}}=\sqrt[{\color{red}{3}}]{8^\color{blue}{2}}=\sqrt[3]{64}=4\small.\)

Ответ: \(\displaystyle 4\small.\)

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