Find the value of 1 squareroot 3i55255Solutionzn 1 sqrt3i5

Find the value of (1 = squareroot 3i)^55/2^55.

Solution

z^n = (1+ sqrt3i)^55/^2^55

in polar form : (a+ib) ^n = r^n (cos(nx) + i*sin(nx) )

r = sqrt( 1+ 3) = 2

angle x = tan^-1(sqrt3/1) = pi/3

(1+ sqrt3i)^55/^2^55 = 2^55{ cos(pi*55/3) + i*sin(pi*55/3) }/ 2^55(cos0 + isin0)

=1{ 1/2 + i*sqrt3/2 }

= 1/2 + i*sqrt3/2