Consider the equation $z^3=c$, where $z\in\mathbb{C}$ and $c$ is a positive constant. One of t

Question

Consider the equation $z^3=c$, where $z\in\mathbb{C}$ and $c$ is a positive constant. One of the solutions of the equation is $z=-2+2i\sqrt3$. Find the value of $c$. Write down the other two solutions of $z^3=c$ giving your answers in the form $a+bi$, where $a,b\in\mathbb{R}$. Find the solutions of $(iz)^3=c$ giving your answers in the form $a+bi$, where $a,b\in\mathbb{R}$. The solutions of $(iz)^3=c$ can be represented on an Argand diagram by three points $A$, $B$ and $C$. Find the area of the triangle $ABC$.

Accepted Answer

(a) $64$ (b) $-2-2i\sqrt3,\ 4$ (c) $2\sqrt3+2i,\ -2\sqrt3+2i,\ -4i$ (d) $12\sqrt3$

Solution:

Handwritten & Typed Solutions