指数の計算
ヒロ
定期テストで出題された問題を解いてみよう。
問題次の計算をせよ。
(1) $27^{\frac{2}{3}}$
(2) $\left(\dfrac{1}{8}\right)^{-\frac{2}{3}}$
(3) $2^{\frac{2}{3}}\times2^{\frac{1}{2}}\div2^{\frac{7}{6}}$
(4) $(8^{\frac{1}{6}}\times8^{\frac{1}{2}})^{\frac{1}{2}}$
(5) $8^{\frac{5}{4}}\div4^{\frac{21}{8}}\times2^{\frac{3}{2}}$
(1) $27^{\frac{2}{3}}$
(2) $\left(\dfrac{1}{8}\right)^{-\frac{2}{3}}$
(3) $2^{\frac{2}{3}}\times2^{\frac{1}{2}}\div2^{\frac{7}{6}}$
(4) $(8^{\frac{1}{6}}\times8^{\frac{1}{2}})^{\frac{1}{2}}$
(5) $8^{\frac{5}{4}}\div4^{\frac{21}{8}}\times2^{\frac{3}{2}}$
【考え方と解答】
(1)
(2)
(1)
\begin{align*}
27^{\frac{2}{3}}&=(3^3)^{\frac{2}{3}} \\[4pt]
&=3^{3\times\frac{2}{3}}=3^2=9
\end{align*}
27^{\frac{2}{3}}&=(3^3)^{\frac{2}{3}} \\[4pt]
&=3^{3\times\frac{2}{3}}=3^2=9
\end{align*}
(2)
\begin{align*}
\left(\dfrac{1}{8}\right)^{-\frac{2}{3}}
&=(2^{-3})^{\frac{2}{3}} \\[4pt]
&=2^{-2}=\dfrac{1}{4}
\end{align*}
(3)\left(\dfrac{1}{8}\right)^{-\frac{2}{3}}
&=(2^{-3})^{\frac{2}{3}} \\[4pt]
&=2^{-2}=\dfrac{1}{4}
\end{align*}
\begin{align*}
&2^{\frac{2}{3}}\times2^{\frac{1}{2}}\div2^{\frac{7}{6}} \\[4pt]
&=2^{\frac{2}{3}+\frac{1}{2}-\frac{7}{6}}=2^{\frac{4+3-7}{6}} \\[4pt]
&=2^0=1
\end{align*}
(4)&2^{\frac{2}{3}}\times2^{\frac{1}{2}}\div2^{\frac{7}{6}} \\[4pt]
&=2^{\frac{2}{3}+\frac{1}{2}-\frac{7}{6}}=2^{\frac{4+3-7}{6}} \\[4pt]
&=2^0=1
\end{align*}
\begin{align*}
&(8^{\frac{1}{6}}\times8^{\frac{1}{2}})^{\frac{1}{2}} \\[4pt]
&=(8^{\frac{1}{6}+\frac{1}{2}})^{\frac{1}{2}} \\[4pt]
&=(8^{\frac{2}{3}})^{\frac{1}{2}}=8^{\frac{1}{3}} \\[4pt]
&=(2^3)^{\frac{1}{3}}=2
\end{align*}
(5)&(8^{\frac{1}{6}}\times8^{\frac{1}{2}})^{\frac{1}{2}} \\[4pt]
&=(8^{\frac{1}{6}+\frac{1}{2}})^{\frac{1}{2}} \\[4pt]
&=(8^{\frac{2}{3}})^{\frac{1}{2}}=8^{\frac{1}{3}} \\[4pt]
&=(2^3)^{\frac{1}{3}}=2
\end{align*}
\begin{align*}
8^{\frac{5}{4}}\div4^{\frac{21}{8}}\times2^{\frac{3}{2}}
&=(2^3)^{\frac{5}{4}}\times(2^2)^{-\frac{21}{8}}\times2^{\frac{3}{2}} \\[4pt]
&=2^{\frac{15}{4}-\frac{21}{4}+\frac{3}{2}} \\[4pt]
&=2^0=1
\end{align*}
8^{\frac{5}{4}}\div4^{\frac{21}{8}}\times2^{\frac{3}{2}}
&=(2^3)^{\frac{5}{4}}\times(2^2)^{-\frac{21}{8}}\times2^{\frac{3}{2}} \\[4pt]
&=2^{\frac{15}{4}-\frac{21}{4}+\frac{3}{2}} \\[4pt]
&=2^0=1
\end{align*}
累乗根の計算
ヒロ
定期テストで出題された問題を解いてみよう。
問題次の計算をせよ。
(1) $\sqrt{6}\times\sqrt[4]{6}\div\sqrt[3]{6^2}$
(2) $\sqrt{5}\times\sqrt[4]{5}\div\sqrt[3]{5^2}$
(3) $-\sqrt[3]{24}+\sqrt[3]{81}-\sqrt[3]{3}$
(4) $\sqrt{a}\times\sqrt[3]{a^2}\div\sqrt[6]{a^5}$
(5) $(a^2\times\sqrt[3]{a^2})^{\frac{1}{4}}$
(1) $\sqrt{6}\times\sqrt[4]{6}\div\sqrt[3]{6^2}$
(2) $\sqrt{5}\times\sqrt[4]{5}\div\sqrt[3]{5^2}$
(3) $-\sqrt[3]{24}+\sqrt[3]{81}-\sqrt[3]{3}$
(4) $\sqrt{a}\times\sqrt[3]{a^2}\div\sqrt[6]{a^5}$
(5) $(a^2\times\sqrt[3]{a^2})^{\frac{1}{4}}$
ヒロ
累乗根の計算では，指数に直すと計算しやすいだろう。
【考え方と解答】
(1)
(1)
\begin{align*}
&\sqrt{6}\times\sqrt[4]{6}\div\sqrt[3]{6^2} \\[4pt]
&=6^{\frac{1}{2}}\times6^{\frac{1}{4}}\div6^{\frac{2}{3}} \\[4pt]
&=6^{\frac{1}{2}+\frac{1}{4}-\frac{2}{3}}=6^{\frac{6+3-8}{12}} \\[4pt]
&=6^{\frac{1}{12}}
\end{align*}
(2)&\sqrt{6}\times\sqrt[4]{6}\div\sqrt[3]{6^2} \\[4pt]
&=6^{\frac{1}{2}}\times6^{\frac{1}{4}}\div6^{\frac{2}{3}} \\[4pt]
&=6^{\frac{1}{2}+\frac{1}{4}-\frac{2}{3}}=6^{\frac{6+3-8}{12}} \\[4pt]
&=6^{\frac{1}{12}}
\end{align*}
\begin{align*}
&\sqrt{5}\times\sqrt[4]{5}\div\sqrt[3]{5^2} \\[4pt]
&=5^{\frac{1}{2}}\times5^{\frac{1}{4}}\div5^{\frac{2}{3}} \\[4pt]
&=5^{\frac{1}{2}+\frac{1}{4}-\frac{2}{3}}=5^{\frac{6+3-8}{12}} \\[4pt]
&=5^{\frac{1}{12}}
\end{align*}
(3)&\sqrt{5}\times\sqrt[4]{5}\div\sqrt[3]{5^2} \\[4pt]
&=5^{\frac{1}{2}}\times5^{\frac{1}{4}}\div5^{\frac{2}{3}} \\[4pt]
&=5^{\frac{1}{2}+\frac{1}{4}-\frac{2}{3}}=5^{\frac{6+3-8}{12}} \\[4pt]
&=5^{\frac{1}{12}}
\end{align*}
\begin{align*}
&-\sqrt[3]{24}+\sqrt[3]{81}-\sqrt[3]{3} \\[4pt]
&=-2\sqrt[3]{3}+3\sqrt[3]{3}-\sqrt[3]{3} \\[4pt]
&=0
\end{align*}
(4)&-\sqrt[3]{24}+\sqrt[3]{81}-\sqrt[3]{3} \\[4pt]
&=-2\sqrt[3]{3}+3\sqrt[3]{3}-\sqrt[3]{3} \\[4pt]
&=0
\end{align*}
\begin{align*}
&\sqrt{a}\times\sqrt[3]{a^2}\div\sqrt[6]{a^5} \\[4pt]
&=a^{\frac{1}{2}}\times a^{\frac{2}{3}}\div a^{\frac{5}{6}} \\[4pt]
&=a^{\frac{1}{2}+\frac{2}{3}-\frac{5}{6}}=a^{\frac{6+8-10}{12}} \\[4pt]
&=a^{\frac{1}{3}}
\end{align*}
(5)&\sqrt{a}\times\sqrt[3]{a^2}\div\sqrt[6]{a^5} \\[4pt]
&=a^{\frac{1}{2}}\times a^{\frac{2}{3}}\div a^{\frac{5}{6}} \\[4pt]
&=a^{\frac{1}{2}+\frac{2}{3}-\frac{5}{6}}=a^{\frac{6+8-10}{12}} \\[4pt]
&=a^{\frac{1}{3}}
\end{align*}
\begin{align*}
&(a^2\times\sqrt[3]{a^2})^{\frac{1}{4}} \\[4pt]
&=(a^2\times a^{\frac{2}{3}})^{\frac{1}{4}} \\[4pt]
&=(a^{2+\frac{2}{3}})^{\frac{1}{4}}=(a^\frac{8}{3})^{\frac{1}{4}} \\[4pt]
&=a^{\frac{2}{3}}
\end{align*}
&(a^2\times\sqrt[3]{a^2})^{\frac{1}{4}} \\[4pt]
&=(a^2\times a^{\frac{2}{3}})^{\frac{1}{4}} \\[4pt]
&=(a^{2+\frac{2}{3}})^{\frac{1}{4}}=(a^\frac{8}{3})^{\frac{1}{4}} \\[4pt]
&=a^{\frac{2}{3}}
\end{align*}