Simplify the following i⁷

Simplify the following
i¹⁹⁴⁷+ i¹⁹⁵⁰

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

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

Write the following in the rectangular form:
\(\overline { \left( 5+9i \right) +\left( 2-4i \right) } \)

If z = x + iy, find the following in rectangular form.
\(Re\left( \frac { 1 }{ z } \right) \)

If z_i = 2− i and z₂ = -4+3i , find the inverse of z₁z₂ and \(\frac { { z }_{ 1 } }{ { z }_{ 2 } } \)

Prove the following properties z is real if and only if z = \(\bar { z } \)

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

Find the following \(\left| \frac { 2+i }{ -1+2i } \right| \) 

Which one of the points i, −2 + i, and 3 is farthest from the origin?

If z₁, z₂ and z₃ are complex numbers such that |z₁| = |z₂| = |z₃| = |z₁+z₂+z₃| = 1 find the value of \(\left| \frac { 1 }{ { z }_{ 1 } } +\frac { 1 }{ z_{ 2 } } +\frac { 1 }{ { z }_{ 3 } } \right| \)

Show that the equation z² = \(\bar { z } \) has four solutions.

Find the square root of 6−8i .

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

If |z| = 3, show that \(7\le \left| z+6-8i \right| \le 13\).

Find the square roots of 4+3i

If z = x + iy is a complex number such that \(\left| \frac { z-4i }{ z+4i } \right| =1\) show that the locus of z is real axis.

Obtain the Cartesian form of the locus of z = x + iy in each of the following cases:
\(\left[ Re\left( iz \right) \right] ^{ 2 }=3\)

Write in polar form of the following complex numbers
\(2+i2\sqrt { 3 } \)

Obtain the Cartesian form of the locus of z in each of the following cases.
|z| = |z - i|

Find the modulus and principal argument of the following complex numbers. 
\(\sqrt { 3 } +i\).

Simplify the following
 i¹⁹⁴⁸ -i ^-1869

Simplify the following
\(\sum _{ n=1 }^{ 12 }{ { i }^{ n } } \)

Simplify the following
\({ i }^{ 59 }+\frac { 1 }{ { i }^{ 59 } } \)

Simplify the following
 i i ²i³...i²⁰⁰⁰

Simplify the following
\(\sum _{ n=1 }^{ 10 }{ { i }^{ n+50 } } \).

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

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

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

Evaluate the following if z = 5−2i and w = −1+3i
z² + 2zw + w²

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

Write the following in the rectangular form:
\(\cfrac { 10-5i }{ 6+2i } \)

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

If z = x + iy, find the following in rectangular form.
Re\(\left( i\bar { z } \right) \)

If z = x + iy , find the following in rectangular form.
Im(3z + 4\(\bar { z } \) − 4i)

Prove the following properties
\(Re\left( z \right) =\frac { z+\bar { z } }{ 2 } \) and Im\(\left( z \right) =\frac { z-\bar { z } }{ 2i } \)

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

Find the modulus of the following complex number \(\frac { 2-i }{ 1+i } +\frac { 1-2i }{ 1-i } \)

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

Find the modulus of the following complex numbers
 2i(3−4i)(4−3i).

Find the square roots of −6+8i

Find the square roots of
−5 −12i .

Obtain the Cartesian form of the locus of z = x + iy in each of the following cases:
 Im[(1−i)z+1] = 0

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

Obtain the Cartesian form of the locus of z = x + iy in each of the following cases:
\(\overline { z } =z^{ -1 }\)

Write in polar form of the following complex numbers
\(3-i\sqrt { 3 } \)

Write in polar form of the following complex numbers
-2 - i2

Write in polar form of the following complex numbers
\(\frac { i-1 }{ cos\frac { \pi }{ 3 } +isin\frac { \pi }{ 3 } } \)

Simplify the following:
i ¹⁷²⁹

Simplify the following:
i ^-1924+ i²⁰¹⁸

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

Simplify the following:
i i²i³...i⁴⁰

Obtain the Cartesian form of the locus of z in in each of the following cases.
|2z - 3 - i| = 3

Find the following \(\left| \overline { (1+i) } (2+3i)(4i-3) \right| \)

Find the following \(\left| \frac { i(2+i)^{ 3 } }{ \left( 1+i \right) ^{ 2 } } \right| \)

Find the modulus and principal argument of the following complex numbers.
\(-\sqrt { 3 } +i\)

Find the modulus and principal argument of the following complex numbers:
\(-\sqrt { 3 } -i\)

Find the modulus and principal argument of the following complex numbers.
\(\sqrt { 3 } \)-i