Given `z=(1+isqrt(3))^(100),` then `[R E(z)//I M(z)]` equals `2^(100)` b. `2^(50)` c. `1/(sqrt(3))` d. `sqrt(3)`

Options

A. `2^(100)`
B. `2^(50)`
C. `(1)/(sqrt(3))`
D. `sqrt(3)`

Answer

C

Explanation

`z=(1+isqrt(3))^(100)=2^(100)(cos""(pi)/(3)+isin""(pi)/(3))^(100)`
`=2^(100)(cos""(100pi)/(3)+isin""(100pi)/(3))`
`=2^(100)(-cos""(pi)/(3)-isin""(pi
)/(3))`
`=2^(100)(-(1)/(2)-(isqrt(3))/(2))`
`rArr (Re(z))/(Im(z))=(-1//2)/(-sqrt(3)//2)=(1)/(sqrt(3))`