Learn with Amar
Teaching video
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Grade 12 chapters and video availability01 · Read and understand
What you will learn
- Apply polar powers while retaining every angular branch.
- Justify the conclusion "The roots are 2, −1+i√3, −1−i√3" using the stated assumptions.
Before you start
Complex polar form and De Moivre's formula.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Find all complex solutions of z³=8.
Why this math matters
Apply polar powers while retaining every angular branch. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- Complex solutions are allowed.
- Angles are considered modulo a full turn.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Find all complex roots using equally spaced angles
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Find all complex solutions of z³=8.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
Write 8=8(cos(2πk)+i sin(2πk))
The positive real number has arguments differing by full turns.
Work through the mathematics
Root magnitude is two and angles are 2πk/3 for k=0,1,2
Dividing the argument by three produces three distinct directions.
Check the conclusion
The roots are 2, −1+i√3, −1−i√3
Each cubes to eight; the three roots lie equally spaced on a radius-two circle.
The result
The roots are 2, −1+i√3, −1−i√3
Each cubes to eight; the three roots lie equally spaced on a radius-two circle.
Common mistakes to catch
- A real cube root is only one of the complex roots.
- Root magnitude is a cube root of the original magnitude.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What is the sum of the three roots?
Show a hint
Add their real and imaginary parts.
Reveal answer and explanation
Zero
Two plus minus one plus minus one cancels, as do the imaginary parts.
Practice 2
Why stop at k=2?
Show a hint
Adding three to k adds a full turn to the root angle.
Reveal answer and explanation
Later values repeat the same roots
There are exactly three distinct angular classes.
Take the idea with you
Use rotational symmetry to organize roots of a complex power equation.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Recognize a circle hidden in a polar equation
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