Surds and Radicals
A radical is an expression that uses the root symbol (√). A surd is a special type of radical — an irrational root that cannot be simplified to a rational number.
For example, √4 = 2 is a radical but not a surd (it simplifies to a rational number). In contrast, √2 = 1.41421... is a radical and a surd because it cannot be simplified further.
In NCERT Class 9 Mathematics, surds arise naturally when studying the Number Systems chapter, operations on real numbers, and rationalisation of denominators.
What is Surds and Radicals?
Definition: A surd is an irrational root of a rational number. If a is a positive rational number and n is a positive integer such that n√a is irrational, then n√a is called a surd.
Conditions for a surd:
- The number under the root (radicand) must be a positive rational number.
- The result must be irrational.
- The root must be a positive integer (≥ 2).
Terminology:
- Radical sign: The symbol √ (or n√)
- Radicand: The number under the radical sign (e.g., 5 in √5)
- Index (or order): The value of n in n√a. For square roots, n = 2 (usually omitted).
Not surds:
- √9 = 3 (rational, so not a surd)
- ³√27 = 3 (rational, so not a surd)
- √(π) — π is not rational, so this is not a surd by definition
Surds and Radicals Formula
Laws of Surds:
1. Multiplication of surds (same order):
n√a × n√b = n√(ab)
2. Division of surds (same order):
n√a ÷ n√b = n√(a/b)
3. Power of a surd:
(n√a)m = n√(am)
4. Rationalising factor:
- The rationalising factor of √a is √a, since √a × √a = a (rational).
- The rationalising factor of a + √b is a − √b, since (a + √b)(a − √b) = a² − b.
5. Changing the order:
- n√a = mn√(am)
- Example: √2 = 4√(2²) = 4√4
Derivation and Proof
Types of Surds:
1. Pure Surd:
- A surd with no rational factor other than 1.
- Examples: √2, √5, ³√7
2. Mixed Surd:
- A surd with a rational coefficient multiplied by a surd.
- Examples: 3√2, 5√3, 2³√5
- A pure surd can be expressed as a mixed surd: √12 = 2√3 (mixed)
3. Like Surds:
- Surds with the same radicand (after simplification).
- Examples: 3√5 and 7√5 are like surds; √12 = 2√3 and √27 = 3√3 are also like surds.
- Like surds can be added and subtracted.
4. Unlike Surds:
- Surds with different radicands that cannot be simplified to the same radicand.
- Examples: √2 and √3 are unlike surds.
- Unlike surds cannot be combined by addition or subtraction.
5. Order of a Surd:
- The index n in n√a is the order of the surd.
- √5 is a surd of order 2 (quadratic surd).
- ³√5 is a surd of order 3 (cubic surd).
- 4√5 is a surd of order 4.
6. Compound Surd:
- An expression involving the sum or difference of two or more surds.
- Examples: √2 + √3, 3 + √5
7. Binomial Surd:
- A compound surd with exactly two terms.
- Examples: 2 + √3, √5 − √2
- The conjugate of a + √b is a − √b.
Types and Properties
Operations on Surds:
1. Addition and Subtraction:
- Only like surds can be added or subtracted.
- 3√5 + 7√5 = 10√5
- 5√3 − 2√3 = 3√3
- √2 + √3 cannot be simplified further (unlike surds).
2. Multiplication:
- √a × √b = √(ab) for surds of the same order.
- 2√3 × 4√3 = 8 × 3 = 24
- √2 × √6 = √12 = 2√3
3. Division:
- √a ÷ √b = √(a/b)
- √18 ÷ √2 = √9 = 3
4. Rationalisation:
- To rationalise 1/√a, multiply numerator and denominator by √a.
- 1/√3 = √3/3
- To rationalise 1/(a + √b), multiply by (a − √b)/(a − √b).
5. Simplification of surds:
- Factor the radicand and extract perfect squares.
- √72 = √(36 × 2) = 6√2
- √50 = √(25 × 2) = 5√2
6. Comparison of surds:
- To compare √a and √b: compare a and b directly.
- To compare surds of different orders, convert to the same order using LCM of indices.
- Compare √3 and ³√5: LCM(2,3) = 6. √3 = 6√27, ³√5 = 6√25. Since 27 > 25, √3 > ³√5.
Solved Examples
Example 1: Example 1: Identify surds
Problem: Which of the following are surds? (i) √16 (ii) √5 (iii) ³√8 (iv) ³√10 (v) √(−4)
Solution:
- (i) √16 = 4 → Rational → Not a surd
- (ii) √5 = 2.2360... → Irrational, radicand is rational → Surd
- (iii) ³√8 = 2 → Rational → Not a surd
- (iv) ³√10 → Irrational, radicand is rational → Surd
- (v) √(−4) → Not a real number → Not a surd
Answer: (ii) √5 and (iv) ³√10 are surds.
Example 2: Example 2: Simplify √72
Problem: Simplify √72.
Solution:
- Find the prime factorisation: 72 = 2³ × 3² = 36 × 2
- √72 = √(36 × 2) = √36 × √2 = 6√2
Answer: √72 = 6√2
Example 3: Example 3: Add surds
Problem: Simplify 3√12 + 2√27 − √48.
Solution:
Simplify each surd:
- 3√12 = 3 × 2√3 = 6√3
- 2√27 = 2 × 3√3 = 6√3
- √48 = 4√3
Combine like surds:
- 6√3 + 6√3 − 4√3 = (6 + 6 − 4)√3 = 8√3
Answer: 8√3
Example 4: Example 4: Multiply surds
Problem: Find the product: √6 × √15.
Solution:
- √6 × √15 = √(6 × 15) = √90
- Simplify: 90 = 9 × 10
- √90 = √9 × √10 = 3√10
Answer: √6 × √15 = 3√10
Example 5: Example 5: Rationalise the denominator
Problem: Rationalise the denominator of 5/√7.
Solution:
- Multiply numerator and denominator by √7.
- 5/√7 = (5 × √7)/(√7 × √7) = 5√7/7
Answer: 5/√7 = 5√7/7
Example 6: Example 6: Rationalise a binomial surd denominator
Problem: Rationalise the denominator of 1/(3 + √2).
Solution:
- Multiply by the conjugate: (3 − √2)/(3 − √2)
- Numerator: 1 × (3 − √2) = 3 − √2
- Denominator: (3 + √2)(3 − √2) = 9 − 2 = 7
Answer: 1/(3 + √2) = (3 − √2)/7
Example 7: Example 7: Convert mixed surd to pure surd
Problem: Express 3√5 as a pure surd.
Solution:
- 3√5 = √(3²) × √5 = √9 × √5 = √(9 × 5) = √45
Answer: 3√5 = √45
Example 8: Example 8: Compare two surds
Problem: Which is greater: √7 or ³√11?
Solution:
Method: Convert both to surds of the same order.
- LCM of orders (2 and 3) = 6.
- √7 = 6√(7³) = 6√343
- ³√11 = 6√(11²) = 6√121
- Compare: 343 > 121
Answer: √7 > ³√11
Example 9: Example 9: Simplify (2 + √3)(2 − √3)
Problem: Simplify (2 + √3)(2 − √3).
Solution:
- Using (a + b)(a − b) = a² − b²:
- = 2² − (√3)²
- = 4 − 3
- = 1
Answer: (2 + √3)(2 − √3) = 1
Example 10: Example 10: Simplify √(5 + 2√6)
Problem: Simplify √(5 + 2√6).
Solution:
Express 5 + 2√6 as a perfect square:
- 5 + 2√6 = 3 + 2 + 2√(3 × 2) = (√3)² + (√2)² + 2 × √3 × √2
- This is of the form (a + b)² = a² + 2ab + b² where a = √3, b = √2.
- So 5 + 2√6 = (√3 + √2)²
- √(5 + 2√6) = √3 + √2
Answer: √(5 + 2√6) = √3 + √2
Real-World Applications
Applications of Surds and Radicals:
- Geometry: The diagonal of a square with side 1 is √2, a surd. Surds appear in lengths of diagonals, heights of equilateral triangles (√3/2 × side), and trigonometric ratios (sin 45° = √2/2).
- Trigonometry: Standard angle values involve surds: sin 30° = 1/2, cos 30° = √3/2, tan 60° = √3. These exact values are essential for precise calculations.
- Engineering: Structural calculations use exact surd values to maintain precision, especially in stress and strain analysis.
- Physics: Formulae involving √(2gh) for free-fall velocity, √(L/g) for pendulum period, and similar expressions produce surd values.
- Architecture: The golden ratio (1 + √5)/2 appears in classical and modern design and is a surd-based expression.
- Computer Science: Algorithm complexity often involves surds — the height of a balanced binary tree with n nodes is approximately log&sub2;(n), which can be irrational.
Key Points to Remember
- A surd is an irrational root of a rational number. Not all radicals are surds.
- √4 = 2 is a radical but not a surd. √5 is both a radical and a surd.
- The order of a surd is its root index: √a has order 2, ³√a has order 3.
- Like surds have the same radicand and can be added/subtracted. Unlike surds cannot.
- Pure surds have coefficient 1 (e.g., √5). Mixed surds have a rational coefficient (e.g., 3√5).
- To simplify a surd, factor the radicand and extract perfect powers: √50 = 5√2.
- Rationalisation removes surds from the denominator by multiplying by the conjugate.
- √a × √b = √(ab) holds for surds of the same order.
- To compare surds of different orders, convert to the same order using LCM of indices.
- The conjugate of a + √b is a − √b. Their product is a² − b (rational).
Practice Problems
- Simplify: √200 + √50 − 3√8
- Rationalise the denominator of 7/(2 − √5).
- Express 5√2 as a pure surd.
- Which is greater: 3√2 or 2√5? Justify your answer.
- Simplify: (√5 + √3)(√5 − √3)
- Convert √3 and ³√4 to surds of the same order and compare them.
- Rationalise: (3 + √7)/(3 − √7)
- Simplify: √(7 − 4√3)
Frequently Asked Questions
Q1. What is the difference between a surd and a radical?
A radical is any expression with a root sign (√, ³√, etc.). A surd is a radical whose value is irrational. All surds are radicals, but not all radicals are surds. For example, √9 = 3 is a radical but not a surd.
Q2. What is the order of a surd?
The order of a surd is its root index (n in <sup>n</sup>√a). √5 is order 2 (quadratic surd), ³√5 is order 3 (cubic surd), <sup>4</sup>√5 is order 4.
Q3. Can we add √2 and √3?
No, not in simplified form. √2 and √3 are unlike surds (different radicands), so √2 + √3 cannot be simplified further. It remains as √2 + √3.
Q4. What is the conjugate of a surd?
The conjugate of a + √b is a − √b. Their product is a² − b, which is rational. Conjugates are used to rationalise denominators.
Q5. Why do we rationalise the denominator?
Rationalising the denominator removes surds from the denominator, giving a standard form. This makes arithmetic easier and is the conventional way to express such fractions in mathematics.
Q6. Is √2 × √3 = √6?
Yes. For surds of the same order (both are square roots), √a × √b = √(ab). So √2 × √3 = √6.
Q7. How do you simplify √48?
Factor 48 = 16 × 3. Then √48 = √16 × √3 = 4√3.
Q8. Are surds covered in CBSE Class 9?
Surds are implicitly covered in Chapter 1 (Number Systems) of CBSE Class 9 under irrational numbers, operations on real numbers, and rationalisation of denominators.
