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

## Theorem

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

Let $t > 0$ be wholly real.

Then:

$\cmod {t^z} = t^{\map \Re z}$

## Proof

 $\ds \cmod {t^z}$ $=$ $\ds \cmod {t^{\map \Re z + i \map \Im z} }$ $\ds$ $=$ $\ds \cmod {t^{\map \Re z} t^{i \map \Im z} }$ Sum of Complex Indices of Real Number $\ds$ $=$ $\ds \cmod {t^{\map \Re z} } \cmod {t^{i \map \Im z} }$ Complex Modulus of Product of Complex Numbers $\ds$ $=$ $\ds \cmod {t^{\map \Re z} }$ Modulus of Exponential of Imaginary Number is One:Corollary $\ds$ $=$ $\ds t^{\map \Re z}$ Power of Positive Real Number is Positive

$\blacksquare$