Algebra
# De Moivre's Theorem

Let $z$ be a complex number such that $z = \sqrt{2}(\cos 7 ^{\circ} + i \sin 7 ^{\circ}).$ Then $z^{8}$ can be expressed as $r( \cos \alpha^{\circ} + i \sin \alpha^{\circ})$, where $r$ is a real number and $0 \leq \alpha \leq 90$. What is the value of $r+\alpha ?$

**Details and assumptions**

$i$ is the imaginary unit, where $i^2=-1$.

The two roots of $f(x) = x^2 + 20x + 200$ can be written in the form $a \pm bi$. What is $b-a,$ assuming $b > 0$?

**Details and assumptions**

$i$ is defined as the imaginary unit, such that $i^2 = -1$.

Let $a+bi$ and $c+di$ be the two roots of the quadratic equation $x^2+25=0,$ where $b>d$. What is the value of $ac-bd$?

**Details and assumptions**

$i$ is the imaginary number that satisfies $i^2 = -1$.