Simplify radicals and rationalise a denominator
Extract perfect-power factors · Combine like radicals · Use a conjugate to rationalise a binomial denominator
Section E1.8 · Extended · Roots and radical expressions
Exam scope and assessment notes
Extended content; not a Core requirement. 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
- Square and cube roots
- Expanding brackets
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
- Extract perfect-power factors
- Combine like radicals
- Use a conjugate to rationalise a binomial denominator
Keep exact forms
For non-negative a and b, √(ab) = √a√b. Extract perfect square factors: √72 = √(36×2) = 6√2. Add or subtract only like radicals: 6√2 + 3√2 = 9√2. In general √(a+b) does not equal √a + √b.
For cube roots, extract perfect cube factors: ∛54 = ∛(27×2) = 3∛2. For real a, √(a²) = |a|, so a negative a requires care.
Remove a radical denominator
Multiply numerator and denominator by the same non-zero factor. For 1/√3, multiply by √3 to get √3/3. For 1/(2+√3), use the conjugate 2−√3: the denominator becomes 4−3 = 1.
Check the original and new denominators are non-zero. Rationalising changes the form, not the value; leave an exact result unless a decimal is requested.
Sources for this lesson
Pause and explain
For a = −3, what is √(a²)?
Worked example
Simplify √50 + √8 and rationalise 2/(2+√3).
Show the worked solution
- √50 = 5√2 and √8 = 2√2, giving 7√2.
- Multiply the fraction by (2−√3)/(2−√3); denominator is 1.
Answer 7√2 and 4 − 2√3.
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.
Simplify √48 − √12.
Give me a hint
Extract the largest square factors.
Compare my reasoning
- √48 = 4√3; √12 = 2√3.
- Subtract coefficients.
2√3.
Look for these in your work
- Use exact factors
- Combine only like radicals
Rationalise 3/√5 and simplify ∛128.
Give me a hint
Use √5/√5 for the fraction and 64×2 for the root.
Compare my reasoning
- 3/√5 = 3√5/5.
- ∛128 = ∛(64×2) = 4∛2.
3√5/5 and 4∛2.
Look for these in your work
- Multiply both numerator and denominator
- Extract a perfect cube
Rationalise 1/(3−√2). Verify that the new denominator is non-zero.
Give me a hint
Use the conjugate 3+√2.
Compare my reasoning
- The denominator becomes 9 − 2 = 7.
- The numerator is 3+√2; 7 is non-zero.
(3+√2)/7.
Look for these in your work
- Use the matching conjugate
- Preserve the original value
Common mistakes
- Do not combine unlike radicals as if they were coefficients.
- √(a²) equals |a|, not always a.
What can you explain now?
For a = −3, what is √(a²)?
Compare with the explanation
3.
The principal square root is non-negative, so it gives |a|.
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Make it stick
For a = −3, what is √(a²)?
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.