Two complex numbers are given by $z=1+\sqrt{3}\,i$ and $w=b-bi$ where $b$ is a positive real n

Question

Two complex numbers are given by $z=1+\sqrt{3}\,i$ and $w=b-bi$ where $b$ is a positive real number. Express $z$ and $w$ in polar form. Hence find $\left(\dfrac{z}{w}\right)^{12}$ in Cartesian form, giving your answer in terms of $b$.

Accepted Answer

(a) $z=2\left(\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}\right),\ w=b\sqrt2\left(\cos\left(-\frac{\pi}{4}\right)+i\sin\left(-\frac{\pi}{4}\right)\right)$ (b) $-\dfrac{64}{b^{12}}$

Power of a quotient in terms of a real parameter

Power of a quotient in terms of a real parameter

Solution:

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