Surds and Their Applications in Class 9 Math – Complete Notes, Examples, MCQs and Worksheet

Surds and Their Applications

Surds are one of the most important topics in class 9 math. They help students understand irrational numbers and simplify complicated mathematical expressions. Surds are widely used in algebra, geometry, trigonometry, engineering, and science. At Nisar Math Academy, students can learn this topic through detailed notes, recorded video lectures, and comprehensive lectures designed according to the class 9 syllabus.

Introduction to Surds

A surd is an irrational root of a rational number. In simple words, when the square root, cube root, or higher root of a number cannot be expressed as a rational number, it is called a surd.

Examples of Surds

√2

√3

√5

√7

∛2

∛5

Non-Surds

√4 = 2

√9 = 3

√25 = 5

Since these roots give rational numbers, they are not surds.

Definition of Surd

A surd is an irrational number represented in radical form and cannot be simplified into a rational number.

Example:

√18 = √(9 × 2)

√18 = 3√2

Here, √2 is a surd.

Types of Surds

Simple Surd

A surd containing only one irrational term.

Examples:

√2

√5

√11

Compound Surd

A surd consisting of two or more surds.

Examples:

√2 + √3

2√5 − √7

Binomial Surd

A compound surd with two terms.

Examples:

√2 + √5

3√7 − √3

Pure Surd

A surd having no rational factor except 1.

Examples:

√2

√7

Mixed Surd

A surd with a rational coefficient.

Examples:

3√2

5√7

Simplification of Surds

To simplify a surd, express the number under the radical as a product of a perfect square and another number.

Example 1

Simplify √72

√72 = √(36 × 2)

√72 = 6√2

Example 2

Simplify √45

√45 = √(9 × 5)

√45 = 3√5

Addition and Subtraction of Surds

Only like surds can be added or subtracted.

Example

3√2 + 5√2

= 8√2

Example

7√5 − 2√5

= 5√5

Multiplication of Surds

Example

√3 × √12

= √36

= 6

Example

2√2 × 3√5

= 6√10

Division of Surds

Example

(6√12)/(3√3)

= 2√4

= 4

Rationalization of Denominators

Rationalization means removing surds from the denominator.

Example

1/√2

Multiply numerator and denominator by √2:

= √2/2

Example

1/(3√5)

= √5/15

Conjugate of a Surd

For a binomial surd, change the sign between the two terms.

Examples:

Conjugate of (√2 + √3) is (√2 − √3)

Conjugate of (5 + √7) is (5 − √7)

Applications of Surds

Surds have many practical applications in mathematics and science.

Geometry

Finding diagonals and distances.

Example:

Diagonal of a square of side 5 cm

= 5√2 cm

Pythagoras Theorem

Many lengths are expressed as surds.

Example:

√13, √29, √41

Engineering

Used in design calculations involving measurements and dimensions.

Physics

Used in formulas involving velocity, energy, and wave motion.

Architecture

Used in structural measurements and construction planning.

Solved Examples

Example 1

Simplify √98

√98 = √(49 × 2)

√98 = 7√2

Example 2

Simplify 2√3 + 5√3

= 7√3

Example 3

Multiply √8 × √2

= √16

= 4

Example 4

Rationalize 3/√5

= (3√5)/5

Key Points to Remember

• Surds are irrational roots.

• Perfect square roots are not surds.

• Only like surds can be added or subtracted.

• Surds can be simplified by extracting perfect square factors.

• Rationalization removes surds from denominators.

• Surds are useful in geometry, engineering, and science.

Short Notes on Surds and Their Applications

• A surd is an irrational root of a rational number.

• Examples: √2, √3, √5.

• Types of surds include simple, compound, pure, and mixed surds.

• Surds are simplified by extracting perfect square factors.

• Like surds can be added and subtracted.

• Surds can be multiplied and divided using radical laws.

• Rationalization removes surds from denominators.

• Surds are widely used in geometry, physics, engineering, and architecture.

MCQs on Surds and Their Applications

1. Which of the following is a surd?

A) √25

B) √49

C) √2

D) √64

Answer: C

2. Simplify √50.

A) 10√5

B) 5√2

C) 2√5

D) 25√2

Answer: B

3. Which is a mixed surd?

A) √3

B) √7

C) 4√5

D) √11

Answer: C

4. Evaluate √4 × √9.

A) 6

B) 13

C) 36

D) 12

Answer: A

5. Simplify 3√2 + 2√2.

A) 5

B) 6√2

C) 5√2

D) √2

Answer: C

6. Rationalize 1/√3.

A) √3

B) √3/3

C) 1/3

D) 3√3

Answer: B

7. Which of the following is not a surd?

A) √2

B) √3

C) √16

D) √5

Answer: C

8. Simplify √81.

A) 8

B) 9

C) √9

D) 18

Answer: B

9. Conjugate of (√5 + √2) is:

A) √5 + √2

B) √5 − √2

C) √2 − √5

D) 5√2

Answer: B

10. Surds are commonly used in:

A) Geometry

B) Physics

C) Engineering

D) All of these

Answer: D

Worksheet / Assignment

Part A: Simplify the Following

1. √48
2. √75
3. √98
4. √108
5. √147

Part B: Addition and Subtraction

6. 2√3 + 5√3
7. 8√5 − 3√5
8. 7√2 + 4√2
9. 10√7 − 6√7
10. 9√11 + 2√11

Part C: Multiplication

11. √6 × √24
12. √8 × √18
13. 2√3 × 4√5
14. 5√2 × 3√7
15. √12 × √3

Part D: Rationalization

16. 1/√2
17. 3/√5
18. 5/√7
19. 2/√3
20. 4/√11

Part E: Word Problems

21. Find the diagonal of a square whose side is 6 cm.
22. Find the diagonal of a square whose side is 8 cm.
23. A rectangular field has length 12 m and width 5 m. Find its diagonal.
24. A ladder reaches a height of 8 m when placed 6 m from a wall. Find the ladder length.
25. Explain two real-life applications of surds.

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