Modulus of Positive Real Number to Complex Power is Positive Real Number to Power of Real Part

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Theorem

Let $z \in \C$ be a complex number.

Let $t > 0$ be wholly real.

Let $t^z$ be $t$ to the power of $z$ defined on its principal branch.


Then:

$\left \vert {t^{z}}\right \vert = t^{\operatorname{Re} \left({z}\right) }$.


Proof

\(\displaystyle \left \vert{ t^z }\right \vert\) \(=\) \(\displaystyle \left \vert{ t^{\operatorname{Re}\left({z}\right)+ i \operatorname{Im}\left({z}\right)} }\right \vert\)
\(\displaystyle \) \(=\) \(\displaystyle \left \vert{ t^{\operatorname{Re}\left({z}\right)} t^{i \operatorname{Im}\left({z}\right)} }\right \vert\) Sum of Complex Indices of Real Number
\(\displaystyle \) \(=\) \(\displaystyle \left \vert{ t^{\operatorname{Re}\left({z}\right)} }\right \vert \left \vert { t^{i \operatorname{Im}\left({z}\right)} }\right \vert\) Modulus of Product
\(\displaystyle \) \(=\) \(\displaystyle \left \vert{ t^{\operatorname{Re}\left({z}\right)} }\right \vert\) Modulus of Exponential of Imaginary Number is One:Corollary
\(\displaystyle \) \(=\) \(\displaystyle t^{\operatorname{Re} \left({z}\right)}\) Power of Positive Real Number is Positive

$\blacksquare$