Simplifying Expressions with Surds
Surds are irrational numbers expressed as roots that cannot be simplified to rational numbers — such as √2, √3, √5, ∛7. An expression involving surds must often be simplified to its simplest form.
Simplifying surds involves reducing the radicand, combining like surds, multiplying and dividing surds, and rationalising the denominator.
These skills are essential for Class 9 Number Systems and appear throughout trigonometry and coordinate geometry.
What is Simplifying Expressions with Surds?
Definition: A surd is a root that cannot be expressed as a rational number.
- √4 = 2 → not a surd (it simplifies to a rational number)
- √5 → a surd (irrational, cannot be simplified further)
- ∛8 = 2 → not a surd
- ∛9 → a surd
Like surds: Surds with the same radicand: 3√5 and 7√5 are like surds. √2 and √3 are unlike surds.
Simplifying Expressions with Surds Formula
Rules for simplifying surds:
- √(ab) = √a × √b (product rule)
- √(a/b) = √a / √b (quotient rule)
- a√n + b√n = (a+b)√n (combining like surds)
- a√n − b√n = (a−b)√n
- (√a)² = a
- √a × √a = a
Rationalising the denominator:
- For 1/√a: Multiply by √a/√a → √a/a
- For 1/(a+√b): Multiply by (a−√b)/(a−√b) → uses conjugate
- For 1/(√a+√b): Multiply by (√a−√b)/(√a−√b) → gives (√a−√b)/(a−b)
Solved Examples
Example 1: Example 1: Simplifying a surd
Problem: Simplify √72.
Solution:
- 72 = 36 × 2
- √72 = √(36 × 2) = √36 × √2 = 6√2
Answer: √72 = 6√2.
Example 2: Example 2: Adding like surds
Problem: Simplify 3√5 + 7√5 − 2√5.
Solution:
- (3 + 7 − 2)√5 = 8√5
Answer: 8√5.
Example 3: Example 3: Adding unlike surds after simplification
Problem: Simplify √12 + √27.
Solution:
- √12 = √(4×3) = 2√3
- √27 = √(9×3) = 3√3
- 2√3 + 3√3 = 5√3
Answer: 5√3.
Example 4: Example 4: Multiplying surds
Problem: Simplify √6 × √15.
Solution:
- √6 × √15 = √(6 × 15) = √90 = √(9 × 10) = 3√10
Answer: 3√10.
Example 5: Example 5: Rationalising a simple denominator
Problem: Rationalise 5/√3.
Solution:
- Multiply by √3/√3: 5√3 / (√3 × √3) = 5√3/3
Answer: 5√3/3.
Example 6: Example 6: Rationalising with conjugate
Problem: Rationalise 1/(√5 + √2).
Solution:
- Multiply by (√5 − √2)/(√5 − √2):
- = (√5 − √2)/[(√5)² − (√2)²]
- = (√5 − √2)/(5 − 2)
- = (√5 − √2)/3
Answer: (√5 − √2)/3.
Example 7: Example 7: Expanding (√a + √b)²
Problem: Expand (√3 + √2)².
Solution:
- = (√3)² + 2(√3)(√2) + (√2)²
- = 3 + 2√6 + 2
- = 5 + 2√6
Answer: 5 + 2√6.
Example 8: Example 8: Dividing surds
Problem: Simplify √50/√2.
Solution:
- √50/√2 = √(50/2) = √25 = 5
Answer: 5.
Example 9: Example 9: Complex rationalisation
Problem: Rationalise 3/(2√3 − √5).
Solution:
- Conjugate = 2√3 + √5
- = 3(2√3 + √5)/[(2√3)² − (√5)²]
- = 3(2√3 + √5)/(12 − 5)
- = 3(2√3 + √5)/7
- = (6√3 + 3√5)/7
Answer: (6√3 + 3√5)/7.
Example 10: Example 10: Simplify a mixed expression
Problem: Simplify 2√18 − 3√8 + √50.
Solution:
- 2√18 = 2 × 3√2 = 6√2
- 3√8 = 3 × 2√2 = 6√2
- √50 = 5√2
- 6√2 − 6√2 + 5√2 = 5√2
Answer: 5√2.
Real-World Applications
Applications:
- Trigonometry: Values like sin 45° = 1/√2 need rationalisation to √2/2.
- Coordinate geometry: Distance formula often gives surd answers.
- Physics: Root expressions in wave equations, mechanics.
- Engineering: Exact calculations without decimal approximation.
Key Points to Remember
- A surd is a root that is irrational: √2, √3, ∛5 etc.
- Simplify by extracting perfect square factors: √72 = 6√2.
- Only like surds (same radicand) can be added or subtracted.
- √a × √b = √(ab). √a / √b = √(a/b).
- To rationalise 1/√a, multiply by √a/√a.
- To rationalise 1/(√a ± √b), multiply by the conjugate (√a ∓ √b).
- (√a + √b)(√a − √b) = a − b (difference of squares).
- Always simplify surds to their simplest form before combining.
Practice Problems
- Simplify √200.
- Simplify 4√3 + 2√12 − √75.
- Rationalise 7/√7.
- Rationalise 2/(√7 − √3).
- Expand (√5 − √3)².
- Simplify √45 × √20.
- Simplify (3 + √2)(3 − √2).
- Rationalise 1/(1 + √2 + √3). (Hint: group and rationalise step by step.)
Frequently Asked Questions
Q1. What is a surd?
A surd is a root (square root, cube root, etc.) that cannot be simplified to a rational number. Examples: √2, √3, ∛5.
Q2. Why do we rationalise the denominator?
Convention in mathematics requires the denominator to be rational. It also makes expressions easier to compare and compute.
Q3. What is a conjugate?
The conjugate of (a + √b) is (a − √b). Multiplying by the conjugate eliminates the surd from the denominator using (a+b)(a−b) = a²−b².
Q4. Can you add √2 and √3?
No. They are unlike surds. √2 + √3 cannot be simplified further. You can only add/subtract like surds (same radicand).
Q5. Is √4 a surd?
No. √4 = 2, which is rational. A surd must be irrational.
Q6. How do you simplify √72?
Find the largest perfect square factor: 72 = 36 × 2. So √72 = √36 × √2 = 6√2.
