Calculate the following using the polar form.
(a) (2 + 2i)^3 (b) (2i)^3 (c) (\sqrt{3} + i)^5
Question 9.1.13: Calculate the following using the polar form. (a) (2 + 2i)^3...
(2 + 2i)^3 = (2 \sqrt{2} e^ {i\pi /4} )^3 = (2 \sqrt{2})^3 e^ {i (3 \pi /4)} = 16 \sqrt{2} \left(\cos (3 \pi /4) + i \sin (3 \pi / 4)\right) = -16 + 16i
( 2i)^3 = (2 e^ {i\pi /2} )^3 = 2^{3} e^ {i (3\pi /2)} = 8 \left(\cos (3 \pi /2) + i \sin (3 \pi / 2)\right) = -8 i
(\sqrt{3} + i )^5 = ( 2 e^ {i\pi /6})^5 = 2^{5} e^ {i 5\pi /6} = 32 \left(\cos (5 \pi /6) + i \sin (5 \pi / 6)\right) = -16 \sqrt{3} + 16 i
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