Question

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

Answer 1

Answer:a=0

b=32

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 [ √(2)\left ( (1)/(√(2))+i(1)/(√(2))\right )\right ]^(10)`

`z=\left ( √(2)\right )^(10)\left ( \cos 45+i\sin 45\right )^(10)`

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

`z=32\left ( e^{(5\pi )/(2)}\right )`

`(5\pi )/(2)=2\pi +(\pi )/(2)`

`z=32\left ( e^{i(\pi )/(2)\right )`

`z=32\left ( \cos (\pi )/(2)+i\sin (\pi )/(2)\right )`

real part
`a=32* \cos (90)=0`

`b=32* \sin (90)=32`