= Solution
Rationalizing the denominator gives
$$
\frac1{1+i\sqrt3}=\frac{1-i\sqrt3}{4}.
$$
Hence its <real part>, <imaginary part>, <complex modulus>, and <principal argument> are respectively
$$
\boxed{\frac14,\quad-\frac{\sqrt3}{4},\quad\frac12,\quad-\frac\pi3.}
$$
The quadrant matters: the <real part> is positive and the <imaginary part> negative. For the exponential expression, <Euler's formula> gives $e^{2+i\pi/4}=e^2(1+i)/\sqrt2$, so the same four quantities are
$$
\boxed{\frac{e^2}{\sqrt2},\quad\frac{e^2}{\sqrt2},\quad e^2,\quad\frac\pi4.}
$$
General <complex arguments> are obtained from each stated <principal argument> by adding $2\pi k$, $k\in\mathbb Z$.
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