Question

The complex numbers p and q are such that |p|=2 and arg (p)= pi/3; |q|=7 and arg (q) = -pi/4.
Find the modulus and argument of (p/q)^5.

Answer 1

Is it that \[(\frac{ p }{ q })^{5}\]
the same as \[\frac{ |p ^{5}| }{ |q ^{5|} }\]

Answer 2

So that I can modulus of p 5 times i.e 2^5 and modulus of 5 times i.e 7^5.
So I get 32/16807

Answer 3

*multiply modulus

Answer 4

I was thinking you could write them as exponentials like this,
\[\Large\rm |p|=2,~Arg(p)=\frac{\pi}{3}\qquad\implies\qquad p=2e^{\mathcal i \pi/3}\]

And then just use simple exponential rules.
Yah that looks right for the coefficients.

Answer 5

\[\Large\rm \left(\frac{p}{q}\right)^5=\left(\frac{2e^{\mathcal i \pi/3}}{7e^{-\mathcal i \pi/4}}\right)^5\]

Answer 6

\[\Large\rm =\frac{2^5}{7^5}e^{5\mathcal i(\pi/3+\pi/4)}\]Something like that, yes? :o

Answer 7

Okay. So the final answer for modulus would be \[\frac{ 2^{5} }{ 7^{5} }\]?
And for argument , it would be \[\frac{ 5\pi }{ 3 } +\frac{ 5\pi }{ 4 }\]

Answer 8

Mmm yah I think we're on the right track here.
Make sure you simplify your argument though :D

Answer 9

combine the fractions and stuff

Answer 10

Yes, I will. Thank you so much. ^_^ I was just confused about the power 5, like how should I work with that.

Answer 11

yah complex is weird and fun :3