PRE CALCULUS Pre-Calculus Complex Numbers 3

Question 1

What is the result of dividing (6 + 2i) by (1 + i)?

Accepted Answer

4 - 2i

Answer

Multiply numerator and denominator by the conjugate (1 - i): (6+2i)(1-i)/[(1+i)(1-i)] = (6-6i+2i-2i²)/2 = (8-4i)/2 = 4 - 2i.

Question 2

Using De Moivre's Theorem, what is [2(cos 30° + i sin 30°)]³?

Accepted Answer

8(cos 90° + i sin 90°)

Answer

By De Moivre's Theorem, r^n(cos nθ + i sin nθ) = 2³(cos 90° + i sin 90°) = 8(cos 90° + i sin 90°) = 8i.

Question 3

How many distinct cube roots does any nonzero complex number have?

Accepted Answer

3

Answer

Every nonzero complex number has exactly n distinct nth roots, so there are 3 cube roots.

Question 4

What is the real part of (3 - 2i)² ?

Accepted Answer

5

Answer

(3 - 2i)² = 9 - 12i + 4i² = 9 - 12i - 4 = 5 - 12i, so the real part is 5.

Question 5

Which property states that the modulus of a product equals the product of the moduli?

Accepted Answer

|z₁ · z₂| = |z₁| · |z₂|

Answer

The multiplicative property of moduli states that |z₁ · z₂| = |z₁| · |z₂|.

PRE CALCULUS Practice Test