Express the given complex number in the form a + ib: (5i) (-3i ÷ 5).

Express the given complex number in the form a + ib: i⁹ + i¹⁹.

Express the given complex number in the form a + ib: i^-39.

Express the given complex number in the form a + ib:  3(7 + i7) + i (7 + i7).

Express the given complex number in the form a + ib: (1 – i) – (–1 + i6).

Express the given complex number in the form a + ib: (  +  ) – (4 + ).

Express the given complex number in the form a + ib: [(  + ) + (4 + )] – (-  + i).

Express the given complex number in the form a + ib: (1 – i)⁴.

Express the given complex number in the form a + ib: (  + 3i)³

Express the given complex number in the form a + ib: (-2 – )³

Find the multiplicative inverse of the complex number 4 – 3i.

Find the multiplicative inverse of the complex number √5 + 3i.

Find the multiplicative inverse of the complex number -i.

Express the following expression in the form of a + ib.

{(3 + i√5) (3 - i√5)} ÷ {(√3 + i√2) - (√3 - i√2)}

Find the modulus and the argument of the complex number z = -1 - √3

Find the modulus and the argument of the complex number z = -√3 + i.

Convert the given complex number in polar form: 1 – i.

Convert the given complex number in polar form: -1 + i.

Convert the given complex number in polar form: -1 – i.

Convert the given complex number in polar form: -3.

Convert the given complex number in polar form: √3 + i.

 

Convert the given complex number in polar form: i.

Solve the equation x² + 3 = 0

Solve the equation 2x² + x + 1 = 0.

Solve the equation x² + 3x + 9 = 0

Solve the equation –x² + x – 2 = 0

Solve the equation x² + 3x + 5 = 0.

Solve the equation x² – x + 2 = 0

Solve the equation √2x² + x + √2 = 0

Solve the equation √3x² - √2x + 3√3 = 0

Solve the equation: x² + x +  = 0.

Solve the equation x² +  + 1 = 0

Evaluate: [i¹⁸ + ( )²⁵]³.

For any two complex numbers z₁ and z₂, prove that

Re (z₁z₂) = Re z₁ Re z₂ – Im z₁ Im z₂

Reduce { – { } to the standard form.

If x – iy = √ {(a - ib) ÷ (c - id)}, prove that (x² + y²)² = {(a² + b²) ÷ (c² + d²)}

Convert the following in the polar form:

(1) (1 + 7i) ÷ (2 - i)²                                        

(2) (1 + 3i) ÷ (1 – 2i)

Solve the equation 3x² – 4x +  = 0

Solve the equation x² – 2x +  = 0

Solve the equation 27x² – 10x + 1 = 0

Solve the equation 21x² – 28x + 10 = 0.

If z₁ = 2 – i, z₂ = 1 + i, find |z₁ + z₂ + 1|÷|z₁ – z₂ + i|.

If a + ib = (x + i)² ÷ (2x² + 1), prove that a² + b² = (x² + 1)² ÷ (2x + 1)²

Let z₁ = 2 – i, z₂ = -2 + i. Find

(1) Re {z₁z₂ ÷ }                                        

(2) Im (1 ÷ z₁ )

Find the modulus and argument of the complex number (1 + 2i) ÷ (1 – 3i).

Find the real numbers x and y if (x – iy) (3 + 5i) is the conjugate of –6 – 24i.

Find the modulus of {(1 + i) ÷ (1 - i)} - {(1 - i) ÷ (1 + i)}.

If (x + iy)³ = u + iv, then show that:  +  = 4(x² – y²).

If α and β are different complex numbers with |β| = 1, then find | (β - α) ÷ (1 - β) |

Find the number of non-zero integral solutions of the equation |1 – i|^x = 2^x

If (a + ib) (c + id) (e + if) (g + ih) = A + iB, then show that:

(a² + b²) (c² + d²) (e² + f²) (g² + h²) = A² + B²

If {(1 + i) ÷ (1 - i)}^m then find the least positive integral value of m.