Given the complex number z = 2 + 3i, represent the complex numbers in Argand diagram z, iz , and z+iz

Find the rectangular form of the complex numbers
\(\left( cos\frac { \pi }{ 6 } +isin\frac { \pi }{ 6 } \right) \left( cos\frac { \pi }{ 12 } +isin\frac { \pi }{ 12 } \right) \)

If \(cos\alpha +cos\beta +cos\gamma =sin\alpha +sin\beta +sin\gamma =0\) then show that 
(i) \(cos3\alpha +cos3\beta +cos3\gamma =3cos(\alpha +\beta +\gamma )\)
(ii) \(sin3\alpha +sin3\beta +sin3\gamma +sin3\gamma =3sin\left( \alpha +\beta +\gamma \right) \)

Find the quotient \(\frac { 2\left( cos\frac { 9\pi }{ 4 } +isin\frac { 9\pi }{ 4 } \right) }{ 4\left( cos\left( \frac { -3\pi }{ 2 } + \right) isin\left( \frac { -3\pi }{ 2 } \right) \right) } \) in rectangular form

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

Find the least value of the positive integer n for which \(\left( \sqrt { 3 } +i \right) ^{ n }\) purely imaginary

Obtain the Cartesian form of the locus of z = x + iy in each of the following cases:
|z + i| = |z - 1|

Find the rectangular form of the complex numbers
\(\frac { cos\frac { \pi }{ 6 } -isin\frac { \pi }{ 6 } }{ 2\left( cos\frac { \pi }{ 3 } +isin\frac { \pi }{ 3 } \right) } \)

If z = 2−2i, find the rotation of z by θ radians in the counter clockwise direction about the origin when \(\theta =\frac { 3\pi }{ 2 } \).

Simplify the following:
i ¹⁷²⁹