De Moivre's Theorem shows that to raise a complex number to the nth power, the absolute value is raised to the nth power and the argument is multiplied by n.

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 \(i^2 = -1\).

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