冪乗の性質

冪乗の性質

\(n\in\mathbb{Z}\)とする。

(1)

\[ \left(\alpha\beta\right)^{\gamma}=\alpha^{\gamma}\beta^{\gamma} \]

\(\alpha,\beta,\alpha\beta\)の指数関数を1価とする場合は等号が成り立たない。

(2)

\[ \alpha^{\beta}\alpha^{\gamma}\ne\alpha^{\beta+\gamma} \]

\(\alpha\)の指数関数を1価とする場合は等号が成り立つ。

(3)

\[ \alpha^{\beta}\alpha^{n}=\alpha^{\beta+n} \]

(4)

\[ \left(\alpha^{\beta}\right)^{\gamma}\ne\alpha^{\beta\gamma} \]

(5)

\[ \left(\alpha^{\beta}\right)^{n}=\alpha^{n\beta} \]

(6)

\[ \left(\alpha^{\frac{1}{n}}\right)^{\beta}=\alpha^{\frac{\beta}{n}} \]

(1)

\begin{align*} \left(\alpha\beta\right)^{\gamma} & =e^{\gamma\log\left(\alpha\beta\right)}\\ & =e^{\gamma\left(\log\alpha+\log\beta\right)}\\ & =e^{\gamma\log\alpha}e^{\gamma\log\beta}\\ & =\alpha^{\gamma}\beta^{\gamma} \end{align*}

1価とする場合

\begin{align*} \left(\alpha\beta\right)^{\gamma} & =e^{\gamma\Log\left(\alpha\beta\right)}\\ & \ne e^{\gamma\left(\Log\alpha+\Log\beta\right)}\\ & =e^{\gamma\Log\alpha}e^{\gamma\Log\beta}\\ & =\alpha^{\gamma}\beta^{\gamma} \end{align*}

(2)

\begin{align*} \alpha^{\beta}\alpha^{\gamma} & =e^{\beta\log\alpha}e^{\gamma\log\alpha}\\ & =e^{\beta\left(\Log\alpha+\log1\right)}e^{\gamma\left(\Log\alpha+\log1\right)}\\ & =e^{\left(\beta+\gamma\right)\Log\alpha+\beta\log1+\gamma\log1}\\ & =e^{\left(\beta+\gamma\right)\left(\Log\alpha+\log1\right)+\beta\log1+\gamma\log1-\left(\beta+\gamma\right)\log1}\\ & =e^{\left(\beta+\gamma\right)\left(\log\alpha\right)+\beta\log1+\gamma\log1-\left(\beta+\gamma\right)\log1}\\ & =\alpha^{\beta+\gamma}e^{\beta\log1+\gamma\log1-\left(\beta+\gamma\right)\log1}\\ & \ne\alpha^{\beta+\gamma} \end{align*}

1価とする場合

\begin{align*} \alpha^{\beta}\alpha^{\gamma} & =e^{\beta\Log\alpha}e^{\gamma\Log\alpha}\\ & =e^{\left(\beta+\gamma\right)\Log\alpha}\\ & =\alpha^{\beta+\gamma} \end{align*}

(3)

\begin{align*} \alpha^{\beta}\alpha^{n} & =\alpha^{\beta+n}e^{\beta\log1+n\log1-\left(\beta+n\right)\log1}\cmt{\text{(2)より}}\\ & =\alpha^{\beta+n}e^{\beta\log1-\beta\log1}\\ & =\alpha^{\beta+n} \end{align*}

(4)

\begin{align*} \left(\alpha^{\beta}\right)^{\gamma} & =e^{\gamma\log\alpha^{\beta}}\\ & =e^{\gamma\left(\beta\log\alpha+\log1\right)}\\ & =e^{\beta\gamma\log\alpha+\gamma\log1}\\ & =\alpha^{\beta\gamma}e^{\gamma\log1}\\ & \ne\alpha^{\beta\gamma} \end{align*}

(5)

\begin{align*} \left(\alpha^{\beta}\right)^{n} & =\alpha^{\beta n}e^{n\log1}\cmt{\text{(4)より}}\\ & =\alpha^{n\beta} \end{align*}

(6)

\begin{align*} \left(\alpha^{\frac{1}{n}}\right)^{\beta} & =\alpha^{\frac{\beta}{n}}e^{\beta\log1}\cmt{\text{(4)より}}\\ & =e^{\frac{\beta}{n}\log\alpha+\beta\log1}\\ & =e^{\beta\left(\frac{1}{n}\log\alpha+\log1\right)}\\ & =e^{\beta\left(\frac{1}{n}\log\alpha\right)}\\ & =\alpha^{\frac{\beta}{n}} \end{align*}

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冪乗の性質

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