Hint
Hint
For part (a), compute the modulus using \(\sqrt{(-1)^2+(\sqrt{3})^2}\) and determine the argument based on the quadrant. For part (b), multiply the moduli and subtract the angles, then convert back to Cartesian form. For part (c), use De Moivre’s theorem and adjust the angle to its principal value.
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Answer
Complete Answer
a) The modulus of z is 2 and its argument is \(2\pi/3\), so \(z = 2\,cis(2\pi/3)\).
b) Multiplying gives \(zw = 2\sqrt{2}\,cis(2\pi/3 – \pi/4) = 2\sqrt{2}\,cis(5\pi/12)\), which converts to Cartesian form as \((\sqrt{6}-\sqrt{2}) + i(\sqrt{6}+\sqrt{2})\).
c) Using De Moivre’s theorem, \(z^4 = (2\,cis(2\pi/3))^4 = 16\,cis(8\pi/3)\). Since \(8\pi/3\) is coterminal with \(2\pi/3\), we have \(z^4 = 16\,cis(2\pi/3) = -8 + 8\sqrt{3}i\).
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