Question

Express in the form A + iB

${\left(\frac{1+2i}{1+i}\right)}^n$ for $n=+4$

• A. $\frac{7}{4}+6i$
• B. $\frac{8}{25}+\frac{6}{25}i$
• C. $\frac{8}{25}-\frac{6}{25}i$
• D. $\frac{6}{25}+\frac{8}{25}i$

Answer 1

The correct option is A $\frac{7}{4}+6i$

Let n=4, we get
$\left(\right(1+2i{)}^2{)}^2$
$=(1-4+4i{)}^2$
$=(-3+4i{)}^2$
$=(9-16-24i)$
$=-(7+24i)$
Also
$\left(\right(1+i{)}^2{)}^2$
$=(1-1+2i{)}^2$
$=-4$
Hence
$z=\frac{7+24i}{4}$
$|z|=\frac{25}{4}$