Calculate square and cube roots in context
Reverse squaring and cubing · Distinguish a principal square root from equation solutions · Choose a physically meaningful length
Section Paper 2 · Historical Extended · non-calculator
Exam scope and assessment notes
Core and Extended foundations. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Powers and multiplication
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Reverse squaring and cubing
- Distinguish a principal square root from equation solutions
- Choose a physically meaningful length
Two root operations
√a denotes the non-negative principal root for a ≥ 0. ∛a is the unique real cube root and may be negative. Squaring −7 and 7 gives the same 49, but cubing preserves their different signs.
The expression √49 equals 7. The equation x² = 49 instead has solutions ±7. If x is a length, the context excludes the negative solution.
Units reverse too
A square's area A gives side √A; a cube's volume V gives edge ∛V. Square centimetres become centimetres when square-rooted; cubic centimetres become centimetres when cube-rooted. Verify by raising the result back to the original power.
Sources for this lesson
Pause and explain
What is ∛(−125)?
Worked example
A square has area 54.76 cm². Find its side.
Show the worked solution
- √54.76 = 7.4 because 7.4² = 54.76.
- A side length is positive and measured in cm.
Answer 7.4 cm.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Calculate √144 and ∛216.
Give me a hint
Check perfect squares and cubes.
Compare my reasoning
- 12² = 144.
- 6³ = 216.
12 and 6.
Look for these in your work
- Use the matching power
- Check by multiplication
Calculate √0.81 and ∛(−64).
Give me a hint
Check the decimal place value and the odd-root sign.
Compare my reasoning
- 0.9² = 0.81.
- (−4)³ = −64.
0.9 and −4.
Look for these in your work
- Keep the sign where appropriate
- Square or cube back
A cube holds 0.008 m³. Find its edge in metres and centimetres.
Give me a hint
Find the cube root before converting length units.
Compare my reasoning
- 0.2³ = 0.008, so the edge is 0.2 m.
- 0.2 m = 20 cm.
0.2 m = 20 cm.
Look for these in your work
- Use volume units correctly
- Choose the positive physical length
Common mistakes
- √49 is not ±7.
- A volume needs a cube root, not a square root.
What can you explain now?
What is ∛(−125)?
Compare with the explanation
−5.
(−5)³ = −125; an odd root can be negative.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
What is ∛(−125)?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.