Surds

Surds are special numbers in mathematics that appear as roots and cannot be simplified into exact decimals or fractions. These irrational numbers are used frequently in algebra, geometry, and higher-level math.

Let’s explore the meaning of surds in mathematics, the different types of surds, surd rules, solved examples, and how to simplify and rationalise them.

 

Table of Contents

• What Are Surds in Mathematics?
• Definition of Surds
• Types of Surds
• Surd Rules
• Surds and Indices
• Solved Examples on Surds
• Practice Questions on Surds
• Real-Life Use of Surds
• FAQs on Surds

 

What Are Surds in Mathematics?

Surds in mathematics refer to irrational numbers that are left in root form because they cannot be simplified into exact whole numbers or terminating decimals.

For example:

• √2 = 1.4142135… (non-terminating and non-repeating)

• √3 = 1.7320508…

• √5 = 2.2360679…
  
  

These numbers are called surds because they cannot be expressed exactly using rational numbers.

 

Definition of Surds

A surd is the root of a positive real number that cannot be simplified into a rational number.

In simple terms:
Surds = √(non-perfect square)

Examples:

• √2 (cannot be simplified) → surd

• √4 = 2 (perfect square) → not a surd

• ∛11 (cube root of 11) → surd
  
  

Types of Surds

Surds are classified based on their structure and simplification ability. Here are the types of surds in mathematics:

1. Simple Surds

Contain a single root term.
Example: √2, √5

2. Pure Surds

Completely irrational with no rational multiplier.
Example: √3

3. Similar Surds

Surds that have the same irrational part.
Example: 2√5, 3√5 are similar surds

4. Mixed Surds

Can be expressed as a product of a rational number and a pure surd.
Example: 5√2

5. Compound Surds

Addition or subtraction of two or more surds.
Example: √3 + √5

6. Binomial Surds

Surds that involve two terms inside one expression.
Example: √7 + √2

 

Surd Rules

To solve expressions involving surds, you must know the six important surd rules. These help in simplifying, rationalising, and combining surds.

Rule 1: Simplifying Surds

To simplify √a, find the greatest square factor.
Example:
√18 = √(9 × 2) = √9 × √2 = 3√2

 Rule 2: Surds in Fraction Form

Split the square root of a fraction.
Example:
√(12/121) = √12 / √121 = 2√3 / 11

Rule 3: Rationalising the Denominator

Multiply numerator and denominator by the denominator’s surd.
Example:
5/√7 = (5×√7)/(√7×√7) = 5√7 / 7

 Rule 4: Addition of Surds

Add similar surds directly.
Example:
5√6 + 4√6 = 9√6

 Rule 5: Rationalising Binomial Surds (Denominator a – b√n)

Multiply numerator and denominator by conjugate (a + b√n).
Example:
1 / (3 – √2) = [1 × (3 + √2)] / [(3 – √2)(3 + √2)]

 Rule 6: Rationalising Binomial Surds (Denominator a + b√n)

Same as Rule 5, use (a – b√n) as the conjugate.

These six surd rules are the core of simplifying and working with surds in mathematics. Use them regularly to master simplification.

 

Surds and Indices

Surds are connected to indices (powers) in mathematics.

For example:
√9 = 9½
∛5 = 5⅓
√(x) = x½

Exponential form of surds:

√a = a½
⁴√a = a¼
a⁻½ = 1 / √a

Knowing indices helps convert surds into exponential expressions, making them easier to simplify and combine.

 

Solved Examples on Surds

Example 1:

Simplify: √28
Solution:
= √(4 × 7) = √4 × √7 = 2√7

 

Example 2:

Multiply: √7 × √2
Solution:
= √(7 × 2) = √14

 

Example 3:

Divide: √10 / √5
Solution:
= √(10/5) = √2

 

Example 4:

Add: √x + 2√x
Solution:
= (1 + 2)√x = 3√x

 

Example 5:

Rationalise: 5 / √3
Solution:
= (5 × √3) / (√3 × √3) = 5√3 / 3

 

Example 6:

Simplify: (√10 + √3)(√10 – √3)
Solution:
= 10 – 3 = 7 (using a² – b² identity)

 

Example 7:

Rationalise: 1 / [(8√11) – (7√5)]
Solution:
Multiply numerator and denominator by conjugate: (8√11 + 7√5)

Final result:
= [(8√11 + 7√5)] / [704 – 245] = (8√11 + 7√5) / 459

 

Example 8:

Find the conjugate of 4√2 – √3
Solution:
Conjugate = 4√2 + √3

 

Practice Questions on Surds

1. Simplify: √50

2. Rationalise: 1 / (2 + √3)

3. Multiply: 2√2 × 3√5

4. Add: 5√7 + 3√7

5. Solve: (√5 + √2)²

6. Divide: √72 / √2

7. Simplify: (4 / √5)

8. Find the conjugate of: 7 – √11
   
   

 

Real-Life Use of Surds

• Geometry: Diagonals and areas of non-square figures

• Physics: Measuring irrational values in quantum mechanics

• Architecture: Designing with irrational dimensions

• Engineering: Structural roots and stress analysis

• Computer Science: Algorithms involving root values

• Finance: Calculations involving irrational ratios
  
  

Related Links :

Square root of 2 : Learn how to simplify square roots just like √2 with easy examples.

Square root : Master the concept of square roots with step-by-step explanations and examples.

 

Freqently Asked Questions on Surds

1. What are surds?

Ans.Surds are root values (like √2) that cannot be simplified into rational numbers or exact decimals.

2. Are surds irrational?

Ans.Yes, surds are irrational and cannot be expressed as fractions or terminating decimals.

3. Is √4 a surd?

Ans.No, √4 = 2 which is a whole number, so it’s not a surd.

4. What is the cube root of 11? Is it a surd?

Ans.Yes. ∛11 = 2.22398... (non-repeating, non-terminating), so it is a surd.

5. What are surd rules?

Ans.Surd rules are simplification techniques to add, subtract, multiply, divide, and rationalise expressions involving surds.

Learn all about surds, types of surds, and how to use surd rules to solve problems like a pro.Start now at Orchids The International School with confidence!