Show that `(sqrt3/2 + i/2)^3 = i`

Show that (-1+sqrt(-3i))^3 is a real number

show that ((sqrt(3)+i)/2)^6+((i-sqrt(3))/2)^6=-2

Show that ((-1+sqrt(3)i)/2)^n+((-1-sqrt(3i))/2)^n is equal to 2 when n is a multiple of 3 and is equal to -1 when n is any other positive integer.

Show that |(2z + 5)(sqrt2-i)|=sqrt(3) |2z+5| , where z is a complex number.

Show that (1+ 2i)/(3+4i) xx (1-2i)/(3-4i) is real

If z= ((sqrt3)/(2) + (i)/(2))^(107) + ((sqrt3)/(2)-(i)/(2))^(107) , then show that Im(z)=0

Let a=2i−j+k,b=i+2j−k and c=i+j−2k be three vectors. A vector in the plane of b and c whose projection on a is of magnitude (sqrt3/2) ​ ​ is

Prove that : (i) sqrt(i)= (1+i)/(sqrt(2)) (ii) sqrt(-i)=(1- i)/(sqrt(2)) (iii) sqrt(i)+sqrt(-i)=sqrt(2)

Express the following in the form a+bi (sqrt3 + 5i) (sqrt3 -5i)^(2) + (-4+ 5i)^(2)

Show that the points representing the complex numbers (3+ 3i), (-3- 3i) and (-3 sqrt3 + 3 sqrt3i) on the Argand plane are the vertices of an equilateral triangle