Level 3 Complex Numbers Walkthrough

2025 NCEA Level 3 Complex Numbers Question 1(d)

2025 Paper

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Question

\[ z^3 + 8m^{27}i = 0 \]

Solve the equation, where \(m\) is a positive real constant.

Write your solution(s) in polar form, in terms of \(m\).

First walkthrough idea

Focus to try first

rewriting a complex number in polar form and finding all three cube roots using arguments.

Step 1

Start with the power

We want \(z^3\) on its own before we use polar form and cube roots.

Show the first step’s working

We want \(z^3\) on its own before we use polar form and cube roots.

Key result

Isolate \(z^3\).

Walkthrough overview

What this question practises

This 2025 walkthrough is part of AS91577 — Apply the algebra of complex numbers in solving problems.

Method: Writing cube roots in polar form.

This is Question 1(d) from the 2025 NCEA Level 3 Complex Numbers paper for AS91577 — Apply the algebra of complex numbers in solving problems. Use the guided hints to practise the method before revealing the full working.

Calc.nz is an independent learning resource. Compare questions, diagrams, and assessment information with the official NZQA resources for AS91577.

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Learning summary

Review the method, not only the answer

This walkthrough helps you practise writing cube roots in polar form. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

List all three distinct roots with the correct angular spacing and in the form the question requests.

Continue practising

← Question 1(c) Question 1(e) →