Rationalising the denominator: Class 9 practice
To remove a surd from the denominator, multiply top and bottom by a factor that makes the denominator rational: by √a for a/√a, or by the conjugate for √a ± √b.. Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Laws of surds
- (x + y)(x − y) = x² − y²
The idea behind rationalising the denominator
To remove a surd from the denominator, multiply top and bottom by a factor that makes the denominator rational: by √a for a/√a, or by the conjugate for √a ± √b.
Step-by-step method
- Pick the rationalising factor (√a or the conjugate).
- Multiply numerator and denominator by it.
- Use (√a + √b)(√a − √b) = a − b.
- Simplify.
Worked example 1 (Basic)
Rationalise: 24/√6
- Multiply by √6/√6
- = 24√6/6
- = 4√6
- Answer: 4√6
Worked example 2 (Standard)
Rationalise: 1/(√6 + √5)
- Multiply by the conjugate (√6 − √5)
- Denominator: 6 − 5 = 1
- = (√6 − √5)/1
- Answer: (√6 − √5)/1
Worked example 3 (Challenge)
Rationalise: 1/(√5 − √2)
- Multiply by the conjugate (√5 + √2)
- Denominator: 5 − 2 = 3
- = (√5 + √2)/3
- Answer: (√5 + √2)/3
Common mistakes and misconceptions
- Multiplying only the denominator
- Writing (√a + √b)² instead of using the conjugate
Practice questions
- (Basic) Rationalise: 2/√2
- (Basic) Rationalise: 6/√3
- (Standard) Rationalise: 1/(√7 + √5)
- (Standard) Rationalise: 1/(√11 + √2)
- (Challenge) Rationalise: 1/(√7 − √2)
- (Challenge) Rationalise: 1/(√11 − √2)
Answer key with full working
- Q1: Multiply by √2/√2 → = 2√2/2 → = √2 → Answer: √2
- Q2: Multiply by √3/√3 → = 6√3/3 → = 2√3 → Answer: 2√3
- Q3: Multiply by the conjugate (√7 − √5) → Denominator: 7 − 5 = 2 → = (√7 − √5)/2 → Answer: (√7 − √5)/2
- Q4: Multiply by the conjugate (√11 − √2) → Denominator: 11 − 2 = 9 → = (√11 − √2)/9 → Answer: (√11 − √2)/9
- Q5: Multiply by the conjugate (√7 + √2) → Denominator: 7 − 2 = 5 → = (√7 + √2)/5 → Answer: (√7 + √2)/5
- Q6: Multiply by the conjugate (√11 + √2) → Denominator: 11 − 2 = 9 → = (√11 + √2)/9 → Answer: (√11 + √2)/9
Syllabus note
This skill is listed under "Number Systems" for Class 9 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is rationalising the denominator in?
"Number Systems", Class 9 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.