Complex Number Representation and Powers

00:00 5 marks

Let \(z = 1 + \sqrt{3}i\).
(a) Express \(z\) in modulus-argument form. [2]
(b) Find \(z^3\), expressing your answer in the form \(a+bi\), where \(a, b \in \mathbb{R}\). [3]

Part a

3 minutes 2 marks

Question

Express \(z\) in modulus-argument form.

Hint Hint

Hint

Determine \(|z|\) and \(\theta\) for \(z = 1 + \sqrt{3}i\).

Answer Answer
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Complete Answer

\(z = 2 \text{cis}(\pi/3)\)

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Part b

2 minutes 3 marks

Question

Find \(z^3\), expressing your answer in the form \(a+bi\).

Hint Hint

Hint

Use De Moivre’s theorem by cubing the modulus and tripling the argument, then convert to rectangular form.

Answer Answer
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Complete Answer

\(z^3 = -8\)

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