(1)

\begin{align*} \pv\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}\\ & =\pv\alpha^{\gamma}\pv\beta^{\gamma} \end{align*}

例

\(\alpha=\beta=-1,\gamma=\frac{1}{2}\)とすると左辺は\(\pv\left(-1\cdot-1\right)^{\frac{1}{2}}=\pv1^{\frac{1}{2}}=\pv1=1\)となるが、右辺は\(\pv\left(-1\right)^{\frac{1}{2}}\pv\left(-1\right)^{\frac{1}{2}}=\pv i\pv i=i^{2}=-1\)となるので、一般的に\(\pv\left(\alpha\beta\right)^{\gamma}\ne\pv\alpha^{\gamma}\pv\beta^{\gamma}\)となる。

(2)

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

(3)

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

例

\(\alpha=-1,\beta=2,\gamma=\frac{1}{2}\)とすると左辺は\(\pv\left(\pv\left(-1\right)^{2}\right)^{\frac{1}{2}}=\pv\left(1\right)^{\frac{1}{2}}=\pv\left(1\right)=1\)となるが、右辺は\(\pv\left(-1\right)^{2\cdot\frac{1}{2}}=\pv\left(-1\right)=-1\)となるので、一般的に\(\pv\left(\pv\alpha^{\beta}\right)^{\gamma}\ne\pv\alpha^{\beta\gamma}\)となる。

(4)

\(\Log\alpha^{x}=x\Log\alpha\)を満たすので、
\begin{align*} \pv\left(\pv\alpha^{x}\right)^{\gamma} & =e^{\gamma\Log\alpha^{x}}\\ & =e^{\gamma x\Log\alpha}\\ & =\pv\alpha^{\gamma x} \end{align*}

(5)

\begin{align*} \pv\left(\pv\alpha^{\beta}\right)^{n} & =e^{n\Log\pv\alpha^{\beta}}\\ & =\exists_{1}m\in\mathbb{Z}\;,\;e^{n\left(\beta\Log\alpha+2\pi im\right)}\\ & =e^{n\beta\Log\alpha}\\ & =\pv\alpha^{n\beta} \end{align*}

(6)

\begin{align*} \pv\left(\pv\alpha^{-1}\right)^{\gamma} & =e^{\gamma\Log\left(\left|\alpha^{-1}\right|e^{i\Arg\alpha^{-1}}\right)}\\ & =e^{\gamma\left(\ln\left|\alpha\right|^{-1}+i\Arg e^{i\Arg\alpha^{-1}}\right)}\\ & =\exists_{1}n\in\mathbb{Z}\;,\;e^{\gamma\left(\ln\left|\alpha\right|^{-1}+i\Arg e^{i\left(-\Arg\alpha+n\Arg1\right)}\right)}\\ & =e^{\gamma\left(\ln\left|\alpha\right|^{-1}+i\Arg e^{-i\Arg\alpha}\right)}\\ & \ne e^{-\gamma\left(\ln\left|\alpha\right|+i\Arg\alpha\right)}\\ & =e^{-\gamma\Log\alpha}\\ & =\pv\alpha^{-\gamma} \end{align*}

例

\(\alpha=-1,\gamma=\frac{1}{2}\)とすると左辺は\(\pv\left(\pv\left(-1\right)^{-1}\right)^{\frac{1}{2}}=\pv\left(-1\right)^{\frac{1}{2}}=i\)となるが、右辺は\(\pv\left(-1\right)^{-\frac{1}{2}}=\pv e^{-\frac{1}{2}\Log\left(-1\right)}=\pv e^{-\frac{\pi}{2}i}=-i\)となるので、一般的に\(\pv\left(\pv\alpha^{-1}\right)^{\gamma}\ne\pv\alpha^{-\gamma}\)となる。

(1)

\begin{align*} \mv\left(\alpha\beta\right)^{\gamma} & =e^{\gamma\log\left(\alpha\beta\right)}\\ & =e^{\gamma\left(\log\alpha+\log\beta\right)}\\ & =e^{\gamma\left(\Log\alpha+\log1+\Log\beta+\log1\right)}\\ & =e^{\gamma\left(\Log\alpha+\Log\beta+\log1\right)}\\ & =e^{\gamma\Log\alpha}e^{\gamma\Log\beta}e^{\gamma\log1}\\ & =\pv\alpha^{\gamma}\pv\beta^{\gamma}\mv1^{\gamma} \end{align*} \begin{align*} \mv\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}\\ & =\mv\alpha^{\gamma}\mv\beta^{\gamma} \end{align*} \begin{align*} \mv\left(\alpha\beta\right)^{\gamma} & =\pv\alpha^{\gamma}\pv\beta^{\gamma}\mv1^{\gamma}\\ & =\pv\alpha^{\gamma}\mv1^{\gamma}\pv\beta^{\gamma}\\ & =\mv\alpha^{\gamma}\pv\beta^{\gamma} \end{align*} 同様に\(\mv\left(\alpha\beta\right)^{\gamma}=\pv\alpha^{\gamma}\mv\beta^{\gamma}\)となる。

(2)

\begin{align*} \mv\alpha^{\beta}\mv\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}\\ & =\pv\alpha^{\beta+\gamma}\mv1^{\beta}\mv1^{\gamma} \end{align*} \begin{align*} \mv\alpha^{\beta}\mv\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)\Log\alpha+\beta\log1+\left(\beta+\gamma\right)\log1}\\ & =e^{\left(\beta+\gamma\right)\log\alpha+\beta\log1}\\ & =\mv\alpha^{\beta+\gamma}\mv1^{\beta} \end{align*} 同様に、\(\mv\alpha^{\beta}\mv\alpha^{\gamma}=\mv\alpha^{\beta+\gamma}\mv1^{\gamma}\)となる。

(3)

\begin{align*} \mv\alpha^{\beta}\mv\alpha^{n} & =\mv\alpha^{\beta+n}\mv1^{n}\cmt{\text{(2)より}}\\ & =\mv\alpha^{\beta+n} \end{align*}

(4)

\begin{align*} \mv\left(\mv\alpha^{\beta}\right)^{\gamma} & =e^{\gamma\log\alpha^{\beta}}\\ & =e^{\gamma\left(\beta\log\alpha+\log1\right)}\\ & =e^{\beta\gamma\log\alpha+\gamma\log1}\\ & =\mv\alpha^{\beta\gamma}\mv1^{\gamma} \end{align*}

(5)

\begin{align*} \mv\left(\mv\alpha^{\beta}\right)^{n} & =\mv\alpha^{\beta n}\mv1^{n}\cmt{\text{(4)より}}\\ & =\mv\alpha^{n\beta} \end{align*}

(6)

\begin{align*} \mv\left(\mv\alpha^{\frac{1}{n}}\right)^{\beta} & =\mv\alpha^{\frac{\beta}{n}}\mv1^{\beta}\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)}\\ & =\mv\alpha^{\frac{\beta}{n}} \end{align*}

(7)

\begin{align*} \mv\left(\alpha^{\beta}\right) & =e^{\beta\log\left(\alpha\right)}\\ & =e^{\beta\left(\Log\left(\alpha\right)+\log1\right)}\\ & =e^{\beta\Log\left(\alpha\right)+\beta\log1}\\ & =\pv\left(\alpha^{\beta}\right)\mv1^{\beta} \end{align*}