Multiplying, Powers and De Moivre's Theorem Worksheet with Answer Key

What's on the worksheet

1. Use the De Moivre theorem.
   \left(3\operatorname{cis}\dfrac{\pi}{12}\right)^4\left(\sqrt{2}\operatorname{cis}\dfrac{5\pi}{8}\right)^6\left(\dfrac{1}{2}\operatorname{cis}\dfrac{3\pi}{10}\right)^5
2. Write each base in polar form, then use the De Moivre theorem.
   \left(\sqrt{3} + i\right)^7\left(-3 - 3\sqrt{3}\,i\right)^4(-1 - i)^{-5}
3. The rules for polar form need r(θ + iθ) with r > 0 and the same angle in both places.
4. Write (1 + i)^10(√(3) - i)^6 as a + bi.
5. Let z = 1 - √(3)\,i.
6. Multiplying by a complex number w stretches every point of the plane about 0 by |w| and turns it about 0 by an argument of w.
7. A student works out (-1 + i)^5 like this: