Question

Write (1 + i)^10 as a complex number of the form a+ ???????????????? for real numbers aand ????????.

Answer 1

Answer:a=0

b=32

Step-by-step explanation:

Given

Complex number $z=\left ( 1+i\right )^{10}$

any complex number is written in the form

$z=re^{i\theta }$

where r=magnitude

so $z=\left [ \sqrt{2}\left ( \frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}}\right )\right ]^{10}$

$z=\left ( \sqrt{2}\right )^{10}\left ( \cos 45+i\sin 45\right )^{10}$

$z=2^5\left ( e^{i\frac{\pi }{4}}\right )^{10}$

$z=32\left ( e^{\frac{5\pi }{2}}\right )$

$\frac{5\pi }{2}=2\pi +\frac{\pi }{2}$

$z=32\left ( e^{i\frac{\pi }{2}\right )$

$z=32\left ( \cos \frac{\pi }{2}+i\sin \frac{\pi }{2}\right )$

real part $a=32\times \cos (90)=0$

$b=32\times \sin (90)=32$