指数法則
【\(0\) や負の整数の指数】
\(a\neq 0\) で、\(n\) が正の整数のとき
\(a^0=1\), \(a^{-n}=\displaystyle\frac{1}{a^n}\)
【指数法則】
\(a\neq 0\), \(b\neq 0\) で、\(m\), \(n\) が整数のとき、
① \(a^ma^n=a^{m+n}\)
①’ \(\displaystyle\frac{a^m}{a^n}=a^{m-n}\)
② \((a^m)^n=a^{mn}\)
③ \((ab)^n=a^nb^n\)
③’ \(\left(\displaystyle\frac{a}{b}\right)^n=\displaystyle\frac{a^n}{b^n}\)
【累乗根の性質】
\(a>0\), \(b>0\) で \(m\), \(n\), \(p\) が正の整数のとき
① \(\sqrt[n]{a}\sqrt[n]{b}=\sqrt[n]{ab}\)
② \(\displaystyle\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}\)
③ \((\sqrt[n]{a})^m=\sqrt[nb]{a^{mp}}\)
④ \(\sqrt[m]{\sqrt[n]{a}}=\sqrt[mn]{a}\)
⑤ \(\sqrt[n]{a^m}=\sqrt[np]{a^{mp}}\)
指数法則 (問題)
(1) \(4^5\times 2^{-8}\div 8^{-2}\)
(2) \((a^{-1})^3\times a^7\div a^2\)
(3) \((a^2b^{-1})^3\div (ab^{-2})^2\)
(4) \(\sqrt[3]{9}\times\sqrt[3]{81}\)
(5) \(\sqrt[3]{5}\div\sqrt[12]{5}\times\sqrt[8]{25}\)
(6) \(\sqrt[3]{54}+\sqrt[3]{-250}-\sqrt[3]{-16}\)
(7) \(\displaystyle\frac{\sqrt[3]{a^4}}{\sqrt{b}}\times\frac{\sqrt[3]{b}}{\sqrt[3]{a^2}}\times\sqrt[3]{a\sqrt{b}}\)
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解説
(1) \(4^5\times 2^{-8}\div 8^{-2}\)
\(=(2^2)^5\times \displaystyle\frac{1}{2^8}\div (2^3)^{-2}\)
\(=(2^2)^5\times \displaystyle\frac{1}{2^8}\div\frac{1}{2^{6}}\)
\(=2^{10}\times\displaystyle\frac{1}{2^8}\div\frac{1}{2^{6}}\)
\(=\displaystyle\frac{2^{10}\times 2^6}{2^8}\)
\(=\displaystyle\frac{2^{16}}{2^8}\)
\(=2^8=256\)
(2) \((a^{-1})^3\times a^7\div a^2\)
\(=a^{-3}\times a^7\times\displaystyle\frac{1}{a^2}\)
\(=\displaystyle\frac{a^4}{a^2}=a^2\)
(3) \((a^2b^{-1})^3\div (ab^{-2})^2\)
\(=a^6b^{-3}\div a^2b^{-4}\)
\(=\displaystyle\frac{a^6}{b^3}\times\displaystyle\frac{b^4}{a^2}\)
\(=\displaystyle\frac{a^6b^4}{a^2b^3}=a^4b\)
(4) \(\sqrt[3]{9}\times\sqrt[3]{81}\)
\(=9^{\frac{1}{3}}\times 81^{\frac{1}{3}}\)
\(=(3^2)^{\frac{1}{3}}\times (3^4)^{\frac{1}{3}}\)
\(=3^{\frac{2}{3}}\times 3^{\frac{4}{3}}\)
\(=3^{\frac{2}{3}+\frac{4}{3}}\)
\(=3^2=9\)
(5) \(\sqrt[3]{5}\div\sqrt[12]{5}\times\sqrt[8]{25}\)
\(=5^{\frac{1}{3}}\div 5^{\frac{1}{12}}\times(5^2)^{\frac{1}{8}}\)
\(=5^{\frac{1}{3}-\frac{1}{12}}\times 5^{\frac{1}{4}}\)
\(=5^{\frac{1}{4}}\times 5^{\frac{1}{4}}\)
\(=5^{\frac{1}{2}}=\sqrt{2}\)
(6) \(\sqrt[3]{54}+\sqrt[3]{-250}-\sqrt[3]{-16}\)
\(=54^{\frac{1}{3}}+(-250)^{\frac{1}{3}}-(-16)^{\frac{1}{3}}\)
\(=(27\cdot 2)^{\frac{1}{3}}+(-125\cdot 2)^{\frac{1}{3}}-(-8\cdot 2)^{\frac{1}{3}}\)
\(=3\cdot 2^{\frac{1}{3}}-5\cdot 2^{\frac{1}{3}}+2\cdot 2^{\frac{1}{3}}\)
\(=3\sqrt[3]{2}-5\sqrt[3]{2}+2\sqrt[3]{2}=0\)
(7) \(\displaystyle\frac{\sqrt[3]{a^4}}{\sqrt{b}}\times\frac{\sqrt[3]{b}}{\sqrt[3]{a^2}}\times\sqrt[3]{a\sqrt{b}}\)
\(=\displaystyle\frac{(a^4)^{\frac{1}{3}}}{b^{\frac{1}{2}}}\times\frac{b^{\frac{1}{3}}}{(a^2)^{\frac{1}{3}}}\times(ab^{\frac{1}{2}})^{\frac{1}{3}}\)
\(=\displaystyle\frac{a^{\frac{4}{3}}}{b^{\frac{1}{2}}}\times\frac{b^{\frac{1}{3}}}{a^{\frac{2}{3}}}\times a^{\frac{1}{3}}b^{\frac{1}{6}}\)
\(=\displaystyle\frac{a^{\frac{4}{3}}\times a^{\frac{1}{3}}}{a^{\frac{2}{3}}}\times\frac{b^{\frac{1}{3}}\times b^{\frac{1}{6}}}{b^{\frac{1}{2}}}\)
\(=a^{\frac{4}{3}+\frac{1}{3}-\frac{2}{3}}\times b^{\frac{1}{3}+\frac{1}{6}-\frac{1}{2}}\)
\(=a^{\frac{4}{3}+\frac{1}{3}-\frac{2}{3}}\times b^{\frac{2}{6}+\frac{1}{6}-\frac{3}{6}}\)
\(=a^{1}b^{0}=a\)
おわりに
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