Answer
$2 \sqrt 2 -2 \sqrt 2 \ i$
Work Step by Step
We know from the unit circle that:
$\cos \dfrac{7 \pi}{4}=\dfrac{\sqrt 2}{2}$ and $\sin \dfrac{7 \pi}{4}=\dfrac{\sqrt 2}{2}$
Thus,, we simplify the given expression as follows:
$4\left[\cos (\dfrac{7 \pi}{4})+i \ \sin (\dfrac{7 \pi}{4})\right]
\\=4\left(\dfrac{\sqrt 2}{2}\right) - 4\left(i \dfrac{\sqrt 2}{2}\right)
\\=2 \sqrt 2 -2 \sqrt 2 \ i$
