Question

Consider the complex number z =1+iV3.2Which of the following complex numbers best approximates z4?Hint: z has a modulus of 2 and an argument of 60Choose 1 answer:4-6.91-8 - 13.928 - 13.9i-4 - 6.9

Answer 1

Answer

B. -8 - 13.9i

Explanation

z = 1 + i√3

Given that z has a modulus of 2 and an argument of 60, then,

z⁴ = (1 + i√3)⁴ Note: √3 = 1.732, i² = 1

Using Pascal triangle

z⁴ = (1 + 1.732I)⁴ = 1(1)⁴(1.732I)⁰ + 4(1)³(1.732I)¹ + 6(1)²(1.732I)² + 4(1)¹(1.732I)³ +1(1)⁰(1.732I)⁴

z⁴ = 1 + 4(1.732i) - 6(3) - 4(5.196i) + 9

z⁴ = 1 + 6.928i - 18 - 20.784i + 9

Grouping the terms, we have

z⁴ = 1 - 18 + 9 + 6.928i - 20.784i

z⁴ = -8 - 13.9i