Chapter 09 — Powers (Part 3): Surds and Their Simplification

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MATHEMATICS HIPHOP • A-LEVEL PURE MATHEMATICS

CHAPTER 09: POWERS

PART 3: SURDS AND THEIR SIMPLIFICATION

Building exact answers through the correct use of irrational roots.

Learning Objectives

By the end of this lesson, you should be able to:

• Explain what a surd is and distinguish it from a rational root.
• Simplify surds by extracting square factors.
• Add and subtract like surds.
• Multiply and divide expressions involving surds.
• Expand brackets containing surds.
• Use conjugates to rationalise denominators.
• Express exact answers in their simplest surd form.

1. Understanding Surds

A surd is an irrational root expressed exactly in root form. It cannot be simplified to a rational number.

For example, √2, √3 and √5 are surds because their values cannot be expressed exactly as fractions of integers.

Examples of roots that are not surds

√4 = 2

√9 = 3

√25 = 5

These are not surds because their roots are rational integers.

Compare the following:

• √16 = 4 — not a surd.
• √18 = 3√2 — a surd.
• √49 = 7 — not a surd.
• √50 = 5√2 — a surd.

2. Basic Laws of Surds

For non-negative real numbers a and b, the following laws are useful:

Product law

√a × √b = √(ab)

Quotient law

√a ÷ √b = √(a/b), where b is positive.

Square of a surd

(√a)² = a

Important warning: A square root does not distribute over addition or subtraction.

In general:

√(a + b) ≠ √a + √b

For example, √(9 + 16) = √25 = 5, whereas √9 + √16 = 3 + 4 = 7.

3. Simplifying Surds

To simplify a surd, express the number inside the root as a product containing the largest possible perfect-square factor.

Remember that:

√(ab) = √a × √b

Worked Example 1

Simplify √72.

Step 1: Identify a perfect-square factor.

72 = 36 × 2

Step 2: Separate the factors.

√72 = √36 × √2

Step 3: Evaluate the perfect square.

√72 = 6√2

Worked Example 2

Simplify √180.

180 = 36 × 5

√180 = √36 × √5

√180 = 6√5

Worked Example 3

Simplify 3√48.

3√48 = 3√(16 × 3)

= 3 × 4√3

= 12√3

Worked Example 4

Simplify √200 − √32.

√200 = √(100 × 2) = 10√2

√32 = √(16 × 2) = 4√2

Therefore:

√200 − √32 = 10√2 − 4√2

= 6√2

4. Adding and Subtracting Surds

Only like surds can be added or subtracted directly. Like surds have the same root part after simplification.

For example, 3√5 and 7√5 are like surds, but √2 and √3 are unlike surds.

Worked Example 5

Simplify 5√3 + 2√3 − 4√3.

Collect the coefficients:

=(5 + 2 − 4)√3

= 3√3

Worked Example 6

Simplify √12 + √27 − √75.

√12 = 2√3

√27 = 3√3

√75 = 5√3

Therefore:

= 2√3 + 3√3 − 5√3

= 0

Remember: Simplify each surd first. Then collect like terms.

5. Multiplication of Surds

When multiplying surds, multiply the numbers outside the roots and multiply the numbers inside the roots.

√a × √b = √(ab)

Worked Example 7

Simplify √6 × √15.

= √90

= √(9 × 10)

= 3√10

Worked Example 8

Simplify 2√3 × 5√6.

= (2 × 5)(√3 × √6)

= 10√18

= 10 × 3√2

= 30√2

6. Division of Surds

To divide surds, divide the coefficients and simplify the resulting root where possible.

Worked Example 9

Simplify √75 ÷ √3.

= √(75/3)

= √25

= 5

Worked Example 10

Simplify 6√20 ÷ 3√5.

= (6/3) × √(20/5)

= 2 × √4

= 4

7. Expanding Brackets Containing Surds

Surds obey the ordinary laws of algebra. Use the distributive law and collect like terms after expanding.

Worked Example 11

Expand and simplify 3(2 + √5).

= 3 × 2 + 3 × √5

= 6 + 3√5

Worked Example 12

Expand and simplify (2 + √3)(4 + √3).

=(2 × 4) + (2 × √3) + (4 × √3) + (√3 × √3)

= 8 + 2√3 + 4√3 + 3

= 11 + 6√3

Answer: 11 + 6√3

Worked Example 13

Expand and simplify (√5 + 2)(√5 − 2).

=(√5)² − 2√5 + 2√5 − 2²

= 5 − 4

= 1

This is an example of the difference of two squares.

8. Conjugates of Surds

The conjugate of a binomial involving a surd is obtained by changing the sign between its two terms.

For example:

• The conjugate of 3 + √2 is 3 − √2.
• The conjugate of 5 − √7 is 5 + √7.

The important identity is:

(a + b)(a − b) = a² − b²

Conjugates are particularly useful when rationalising denominators.

9. Rationalising a Denominator

Rationalising a denominator means rewriting a fraction so that there is no surd in its denominator.

Case 1: A single surd in the denominator

Worked Example 14

Rationalise 5/√3.

Multiply the numerator and denominator by √3:

= (5/√3) × (√3/√3)

= 5√3/3

Answer: 5√3/3

Case 2: A binomial denominator

When the denominator contains two terms, multiply by its conjugate.

Worked Example 15

Rationalise 4/(2 + √3).

The conjugate of 2 + √3 is 2 − √3.

Multiply the numerator and denominator by this conjugate:

= [4(2 − √3)]/[(2 + √3)(2 − √3)]

= (8 − 4√3)/(4 − 3)

= 8 − 4√3

Worked Example 16

Rationalise 3/(√5 − 1).

The conjugate is √5 + 1.

= [3(√5 + 1)]/[(√5 − 1)(√5 + 1)]

= 3(√5 + 1)/(5 − 1)

= 3(√5 + 1)/4

10. Expressing Answers in the Form a + b√c

Some examination questions require an expression to be written in the form a + b√c, where a and b are rational numbers and c is a positive integer with no square factor greater than 1.

Worked Example 17

Express (2 + √3)/(2 − √3) in the form a + b√3.

Multiply by the conjugate of the denominator:

= [(2 + √3)(2 + √3)]/[(2 − √3)(2 + √3)]

Numerator:

(2 + √3)² = 4 + 4√3 + 3 = 7 + 4√3

Denominator:

4 − 3 = 1

Answer: 7 + 4√3

Hence a = 7 and b = 4.

11. Common Examination Errors

Error 1: Adding unlike surds

Incorrect: √2 + √3 = √5

Correct: √2 + √3 remains √2 + √3.

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Error 2: Splitting a root over addition

Incorrect: √(a + b) = √a + √b

There is no such general law.

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Error 3: Forgetting to simplify

√48 should be written as 4√3, not left as √48.

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Error 4: Multiplying only the denominator when rationalising

Always multiply both numerator and denominator by the same expression.

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Error 5: Forgetting the conjugate identity

(a + b)(a − b) = a² − b², not a² + b².

12. Examination Practice

Attempt the following questions before opening the solutions. Give exact answers in their simplest form.

Exercise A: Simplification

1. Simplify √108.
2. Simplify √245.
3. Simplify 4√75.
4. Simplify 2√18 + √50.
5. Simplify 3√80 − 2√45.

Open worked solutions — Exercise A

1. √108 = √(36 × 3) = 6√3.

2. √245 = √(49 × 5) = 7√5.

3. 4√75 = 4 × 5√3 = 20√3.

4. 2√18 + √50 = 2(3√2) + 5√2 = 11√2.

5. 3√80 − 2√45 = 3(4√5) − 2(3√5) = 6√5.

Exercise B: Operations

1. Simplify √12 × √27.
2. Simplify 3√5 × 2√10.
3. Simplify √147 ÷ √3.
4. Expand and simplify (3 + √2)(2 + √2).
5. Expand and simplify (√7 + 3)(√7 − 3).

Open worked solutions — Exercise B

1. √12 × √27 = √324 = 18.

2. 3√5 × 2√10 = 6√50 = 30√2.

3. √147 ÷ √3 = √49 = 7.

4. (3 + √2)(2 + √2) = 6 + 3√2 + 2√2 + 2 = 8 + 5√2.

5. (√7 + 3)(√7 − 3) = 7 − 9 = −2.

Exercise C: Rationalising Denominators

1. Rationalise 7/√2.
2. Rationalise 5/√3.
3. Rationalise 2/(3 + √5).
4. Rationalise 4/(√7 − 1).
5. Express (1 + √2)/(1 − √2) in the form a + b√2.

Open worked solutions — Exercise C

1. (7/√2) × (√2/√2) = 7√2/2.

2. (5/√3) × (√3/√3) = 5√3/3.

3. Multiply by (3 − √5):

2(3 − √5)/(9 − 5) = (3 − √5)/2.

4. Multiply by (√7 + 1):

4(√7 + 1)/(7 − 1) = 2(√7 + 1)/3.

5. Multiply by (1 + √2):

(1 + √2)²/(1 − 2) = (3 + 2√2)/(−1).

Answer: −3 − 2√2. Thus a = −3 and b = −2.

Exercise D: Mixed Examination Questions

1. Simplify √192 − 2√27 + √75.
2. Evaluate (√3 + √2)².
3. Express 6/(2 − √3) in the form a + b√3.
4. Simplify (√5 + √2)(√5 − √2) + √45.
5. Given that x = √12 + √27, express x in its simplest surd form.

Open worked solutions — Exercise D

1. √192 = 8√3, 2√27 = 6√3 and √75 = 5√3.

Therefore 8√3 − 6√3 + 5√3 = 7√3.

2. (√3 + √2)² = 3 + 2√6 + 2 = 5 + 2√6.

3. Multiply by (2 + √3):

6(2 + √3)/(4 − 3) = 12 + 6√3.

4. (√5 + √2)(√5 − √2) = 5 − 2 = 3.

√45 = 3√5. Answer: 3 + 3√5.

5. √12 + √27 = 2√3 + 3√3 = 5√3.

13. Lesson Summary

• A surd is an irrational root expressed exactly in root form.
• Extract perfect-square factors to simplify surds.
• Only like surds can be added or subtracted.
• Multiply and divide surds using the ordinary laws of roots.
• Expand brackets using the distributive law.
• Use conjugates to rationalise binomial denominators.
• Always simplify the final answer fully.

MATHEMATICS HIPHOP

Making Mathematics Easier, One Concept at a Time.

Keep practising. Understand every step. Build mathematical confidence.

Coming Next: Chapter 09 — Powers, Part 4: Further Applications of Surds and Rationalisation.

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