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.
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
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.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
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.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs contains 21; Coordinate geometry contains 10 teaching lessons and a mixed checkpoint. Nine earlier overviews support selected topics across the syllabus. The other six syllabus areas, full cumulative assessment and human teacher review are not yet complete. Written lesson practice is self-checked, not automatically graded. The checkpoint samples skills and does not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources