How to calculate Polar coordinates for Complex Polynomials of Higher Degree?

Question

When such I have a complex number such as $3 - 4i$,

I can calculate the $r$ with $r=\sqrt{X^2+Y^2} = \sqrt{3^2+4^2}$.

But how do I solve this when I have a complex number such as $(2+6i)^6$

Answer 1

You can use the fact that: $$|\rm\, X^{n}|=|\,X|^{\,n}\quad\color{gray}\forall\text{ strictly positive integers.} $$ And so: $$\left|(2+6i)^6\right|=|2+6i|^{\,6}$$ Since: $$|2+6i|=\sqrt{2^2+6^{2}}=\sqrt{4+36}=\sqrt{40}=2\sqrt{10}$$ it follows that: $$\left|(2+6i)^6\right|=(2\sqrt{10})^6=40^3.$$ If you don't have a number of the form $\rm X^n$, then you may prefer to use the following property: $$|\rm\, X|=\sqrt{\displaystyle\rm \overset{}X\cdot X}$$