Modulus of exponent raised to a real number / complex number

Question

How do I go about finding the modulus of a complex number of the form: $$ z=e^{(\frac{a}{b+ic})} $$ Where a, b, c are real numbers. If the complex number $b+ic$ wasn't in the denominator then it would be trivial using Euler's formula but I just can't wrap my head around how we would find the modulus for said complex number. Any help would be appreciated.

Furthermore, in one my textbooks the modulus squared appeared as being able to be calculated as:

$$|z|^2 = e^{(\frac{a}{b+ic})}*e^{(\frac{a}{b-ic})}$$

Is the complex conjugate of $z$ simply $z$ but using the complex conjugate of the denominator in the exponent instead? Why?

Answer 1

We can write $\frac{a}{b+ic}=\frac{ab-iac}{b^2+c^2}=x+iy$, where $x=\frac{ab}{a^2+b^2}$ and $y=-\frac{ac}{b^2+c^2}$ are real. Then $e^{x+iy}=e^x(\cos y+i\sin y)$ has modulus $e^x$.