Simplify \(\left( \frac { 1+i }{ 1-i } \right) ^{ 3 }-\left( \frac { 1-i }{ 1+i } \right) ^{ 3 }\) into rectangular form

If z₁= 3 - 2i and z₂ = 6 + 4i, find \(\frac { { z }_{ 1 } }{ z_{ 2 } } \) in the rectangular form.

Find the modulus of the following complex numbers
\(\frac { 2i }{ 3+4i } \)

Find the square roots of 4+3i

Show that the following equations represent a circle, and find its centre and radius \(\left| z-2-i \right| =3\)

Find the principal argument Arg z, when z = \(\frac { -2 }{ 1+i\sqrt { 3 } } \)

Evaluate the following if z = 5−2i and w = −1+3i
z − iw

Write the following in the rectangular form:
 \(\overline { 3i } +\frac { 1 }{ 2-i } \).

Find the modulus of the following complex numbers
(1-i)¹⁰

Show that the following equations represent a circle, and, find its centre and radius
|3z-6+12i| = 8

Simplify the following:
 \(\sum _{ n=1 }^{ 102 }{ { i }^{ n } } \)

Represent the complex numbe \(1+i\sqrt { 3 } \) in polar form.

If (cosθ + i sinθ)² = x + iy, then show that x²+y² =1

Find the value of the complex number (i²⁵)³.

Find the values of the real number x and y if 3x + (2x - 3y) i = 6 + 3i⁹.