17. Graphing Complex Numbers
Powers of Complex Numbers (DeMoivre's Theorem)
Multiple Choice
Given , calculate .
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Verified step by step guidance1
Understand that the expression z = 3CiS(\(\frac{4\pi}{5}\)) represents a complex number in polar form, where 3 is the magnitude and \(\frac{4\pi}{5}\) is the angle in radians.
Recall that when raising a complex number in polar form to a power, you raise the magnitude to that power and multiply the angle by the power. This is based on De Moivre's Theorem.
Apply De Moivre's Theorem: To find z^5, calculate the magnitude as 3^5 and the angle as 5 \(\times\) \(\frac{4\pi}{5}\).
Calculate the new magnitude: 3^5 = 243.
Calculate the new angle: 5 \(\times\) \(\frac{4\pi}{5}\) = 4\(\pi\). Therefore, z^5 = 243CiS(4\(\pi\)).
