Simplify surds and rationalise denominators
Keep exact irrational roots in surd form, simplify them by extracting square factors and remove a surd from a denominator.
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which is a simplified form of √50?
What you will learn
- Simplify square roots by extracting perfect-square factors
- Combine like surds and multiply surds
- Rationalise a denominator using a suitable surd factor or conjugate
Extract square factors without changing the exact value
If an integer has a square factor, √(ab) = √a√b lets you take its square root outside the radical. For example, √72 = √(36 × 2) = 6√2.
Only like surds can be combined: 3√2 + 5√2 = 8√2, while √2 + √3 cannot be combined. Multiply surds by multiplying their radicands and simplify afterwards.
Remove a surd from the denominator
For a denominator √a, multiply top and bottom by √a. For a denominator with two terms, multiply by the conjugate so the denominator becomes a difference of squares.
Rationalising changes the form, not the value. Keep the denominator non-zero and simplify the resulting fraction fully.
Pause and explain
What should multiply the numerator and denominator of 3/√5 to rationalise it?
Worked example
Simplify √72 and rationalise 5/√3.
Show the worked solution
- Factor 72 as 36 × 2, where 36 is a perfect square.
- √72 = 6√2.
- Multiply 5/√3 by √3/√3 to obtain 5√3/3.
Answer √72 = 6√2; 5/√3 = 5√3/3.
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 2√12 + √27.
Give me a hint
Simplify each surd first, then combine only like terms.
Compare my reasoning
- √12 = 2√3, so 2√12 = 4√3.
- √27 = 3√3.
- Add the coefficients: 4√3 + 3√3.
7√3.
Look for these in your work
- Each radical simplified
- Like surds identified
- Coefficients combined
Rationalise 6/(2 + √3).
Give me a hint
Multiply by the conjugate 2 − √3 and simplify the difference of squares.
Compare my reasoning
- Multiply top and bottom by 2 − √3.
- The denominator is 2² − (√3)² = 4 − 3 = 1.
- The numerator is 6(2 − √3).
12 − 6√3.
Look for these in your work
- Conjugate selected
- Denominator rationalised
- Numerator simplified
A square has area 98 cm². Write its side length exactly, then find the exact diagonal length.
Give me a hint
Use the positive square root for the side and multiply the side by √2 for the diagonal.
Compare my reasoning
- Side = √98 = √(49 × 2) = 7√2 cm.
- Diagonal of a square is side × √2.
- 7√2 × √2 = 14 cm.
Side 7√2 cm; diagonal 14 cm.
Look for these in your work
- Area converted to a side
- Positive length used
- Diagonal simplified exactly
Common mistakes
- Adding unlike surds
- Taking a square root of each term in a sum
- Multiplying only the denominator when rationalising
- Replacing an exact surd with a rounded decimal when exact form is required
What can you explain now?
Simplify √200.
Compare with the explanation
10√2
200 = 100 × 2, and √100 = 10.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How do you simplify a surd, combine like surds and rationalise a denominator without changing the value?
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.