1. ## complex number help

The complex number $\displaystyle z=\frac{1+i}{3-2i}$.Write down the argument for $\displaystyle z^2$ and find the exact value of |z^2| .

My working :

$\displaystyle z=\frac{1}{13}+\frac{5}{13}i$
$\displaystyle z^2=(\frac{1}{13}+\frac{5}{13}i)^2$
$\displaystyle z^2=\frac{-24+10i}{169}$

Up to here , did i make any mistake . My final answer is wrong so i am wondering where my mistake is . Can someone pls check for me ? Thanks .

Given: .$\displaystyle z\:=\:\frac{1+i}{3-2i}$.

Find the argument for $\displaystyle z^2$ and find the exact value of $\displaystyle |z^2| .$

My working:

$\displaystyle z\:=\:\frac{1}{13}+\frac{5}{13}i$

$\displaystyle z^2\:=\:\left(\frac{1}{13}+\frac{5}{13}i\right)^2 \;=\;\frac{-24+10i}{169}$

Up to here, did i make any mistakes? .
. . . no

My final answer is wrong. .
You could have shown it to us.

We have: .$\displaystyle z^2 \;=\;-\frac{24}{169} + \frac{10}{169}i$

The argument $\displaystyle \theta$ is given by: .$\displaystyle \tan\theta \;=\;\frac{\frac{10}{169}}{-\frac{24}{169}} \;=\;-\frac{5}{12}$

Hence: .$\displaystyle \theta \;=\;\tan^{-1}\left(\text{-}\tfrac{5}{12}\right) \;\approx\; 2.7468\text{ radians} \;\approx\;157.38^o$

The magnitude of $\displaystyle z^2$ is: .$\displaystyle |z^2| \;=\;\sqrt{\left(\frac{\text{-}24}{169}\right)^2 + \left(\frac{10}{169}\right)^2}\;= \;\sqrt{\frac{676}{169^2}}$

Therefore: .$\displaystyle |z^2| \;=\;\frac{26}{169} \;=\;\frac{2}{13}$