Surds and rationalising a denominator
Keep irrational values exact, simplify square-root expressions and remove roots from a denominator without changing a fraction.
Extended only. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Find square factors and square roots.
- Expand brackets and preserve a fraction when multiplying both parts.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which simplification of √72 is correct?
What you will learn
- Simplify square roots using square factors.
- Combine, multiply and expand surd expressions.
- Rationalise simple and two-term denominators.
Remove complete square factors
A surd is an irrational root left in exact form. For √72, write 72 = 36 × 2 and use √72 = √36 × √2 = 6√2. This product rule holds for non-negative inputs. Choose the largest convenient square factor or simplify in stages until no square factor remains inside the root.
A rounded decimal such as 8.49 approximates 6√2 and is not equal to it exactly. Also, not every root is a surd: √49 = 7 is rational. The square-root symbol gives the non-negative principal root. Do not replace √(a + b) by √a + √b; for example, √(9 + 16) is 5, not 7.
Only matching roots combine by addition
After simplifying, combine like surd terms: √50 + √8 = 5√2 + 2√2 = 7√2. The coefficients add because the common quantity is √2. Terms such as √2 + √3 have different roots and cannot be combined into a single term by adding what is inside them.
For multiplication, multiply coefficients and roots: (2√3)(3√6) = 6√18 = 18√2. Expanding (√3 + 2)² gives 3 + 4√3 + 4 = 7 + 4√3. The middle term matters; squaring a sum does not mean squaring its two terms and omitting their cross product.
Multiply the whole fraction by one
To rationalise 5/√3, multiply numerator and denominator by √3. The value stays the same because the factor is √3/√3 = 1, and the denominator becomes three. The result is 5√3/3. Multiplying only the denominator would change the original fraction.
If the denominator is 2√5, multiply both parts by √5 and then simplify: 3/(2√5) = 3√5/10. The purpose is an equivalent exact form with a rational denominator, rather than a numerical approximation. Check equivalence by multiplying back or comparing a sufficiently precise decimal.
Two-term denominators use a conjugate
The conjugate of 3 + √2 is 3 − √2. Their product is (3 + √2)(3 − √2) = 9 − 2 = 7 because the two cross terms cancel. Thus 1/(3 + √2) = (3 − √2)/7. Choose the sign change on the complete second term and preserve both numerator and denominator.
This uses the difference-of-squares identity (a + b)(a − b) = a² − b². The original denominator and the resulting denominator must be non-zero. For a difference such as 1/(√5 − 1), multiply by √5 + 1 to get (√5 + 1)/4. Keep the exact root until the question specifically asks for a decimal answer.
Pause and explain
Which factor rationalises the denominator 3 + √2?
Worked example
Simplify √200 − √32, then rationalise 3/√2.
Show the worked solution
- √200 = 10√2 and √32 = 4√2, so their difference is 6√2.
- Multiply 3/√2 by √2/√2.
- The numerator is 3√2 and denominator is 2, giving 3√2/2.
Answer 6√2; 3√2/2
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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 √50 + √8 − √18.
Give me a hint
Express each term as a multiple of √2.
Compare my reasoning
- √50 = 5√2, √8 = 2√2 and √18 = 3√2.
- Combine coefficients: 5 + 2 − 3 = 4.
- The simplified exact answer is 4√2.
4√2
Look for these in your work
- Removed square factors before combining.
- Kept the subtraction sign with the final term.
Simplify (√3 + 2)² and rationalise 5/√3.
Give me a hint
Use all terms in the expansion; multiply both parts of the fraction by √3.
Compare my reasoning
- The square expands to 3 + 4√3 + 4 = 7 + 4√3.
- 5/√3 = 5√3/(√3 × √3).
- The denominator is three, so the rationalised form is 5√3/3.
7 + 4√3; 5√3/3
Look for these in your work
- Included the middle term in the square.
- Changed numerator and denominator by the same factor.
Rationalise 1/(3 + √2) and 2/(√5 − 1).
Give me a hint
Use the conjugate of each denominator.
Compare my reasoning
- Multiply the first fraction by (3 − √2)/(3 − √2), giving (3 − √2)/7.
- Multiply the second by (√5 + 1)/(√5 + 1), giving 2(√5 + 1)/(5 − 1).
- Simplify the second to (√5 + 1)/2.
(3 − √2)/7; (√5 + 1)/2
Look for these in your work
- Used a difference of squares for each denominator.
- Simplified the resulting rational coefficient.
A square has exact area 72 cm². Find its exact side and perimeter, then show that multiplying your side by 6√2 reproduces the area.
Give me a hint
Use a root for the side and simplify before finding the perimeter.
Compare my reasoning
- Side = √72 = 6√2 cm.
- Perimeter = 4 × 6√2 = 24√2 cm.
- Check area: (6√2)(6√2) = 36 × 2 = 72 cm².
Side 6√2 cm; perimeter 24√2 cm.
Look for these in your work
- Kept the requested exact form.
- Distinguished linear perimeter units from area units.
Common mistakes
- Splitting the square root of a sum into a sum of roots.
- Combining unlike surd terms.
- Changing only the denominator when rationalising.
- Omitting the middle term when squaring a two-term expression.
What can you explain now?
Simplify 2√12 + √27.
Compare with the explanation
7√3
2√12 = 4√3 and √27 = 3√3, so add their matching coefficients.
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Make it stick
Simplify √98 and rationalise 1/(2 + √3) without notes. Explain why multiplying by a conjugate removes the root from the denominator.
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.