EDEXCEL INTERNATIONAL GCSE MATHEMATICS

Surds

STUDENT WORKBOOK

Contents
1.1 Introduction to Surds ....................................................................................................... 2
1.2 Multiplication of surds ..................................................................................................... 2
1.2.1 Squaring surds........................................................................................................... 2
1.3 Division of Surds ............................................................................................................... 3
1.4 Simplifying surds .............................................................................................................. 3
1.5 Addition and subtraction of surds.................................................................................... 4
1.6 Expanding brackets containing surds ............................................................................... 5
1.7 Rationalizing the denominator......................................................................................... 7

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1.1 INTRODUCTION TO SURDS
A surd is any number that contains a square root sign.

Examples: √3, √12, 3√2

We use them because they are more accurate and you don’t need calculators when working
with them.
Consider the question below, find the exact value of 𝑥:

1.2 MULTIPLICATION OF SURDS

Consider the following:

√4 × √9 = 2 × 3 = 6 √16 × √25 = 4 × 5 = 20

√4 × √9 = √4 × 9 = √36 = 6 √16 × √25 = √16 × 25 = √400 = 20

In general, it follows that √𝑎 × √𝑏 = √𝑎 × 𝑏 = √𝑎𝑏

Evaluate the following (non-calculator):

a) √3 × √7 b) √2 × √5 c) √3 × √17

d) 2√3 × √7 e) 3√2 × 2√5 f) 4√3 × 5√7

1.2.1 SQUARING SURDS

2
√𝑎 × √𝑎 = (√𝑎) = 𝑎
Evaluate the following:

2 2 2
a) (√7) b) (√5) c) (√17)
2 2 2
d) (2√3) e) (3√2) f) (5√7)
4 3
g) 3√2 × 5√2 h) (√2) i) (5√2)

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1.3 DIVISION OF SURDS
Consider the following:

√16 4 √100 10
= =2 = =2
√4 2 √25 5

√16 16 √100 100

= √ = √4 = 2 =√ = √4 = 2
√4 4 √25 25

√𝑎 𝑎
In general, it follows that =√
√𝑏 𝑏

Simplify:

√20 3√15
a) b) 6√18
√4 √3 c)
2√3

7√3 √54 √147

d) e) f)
√21 7√2 49√3

1.4 SIMPLIFYING SURDS

To simplify a surd, write the number under the square root sign as the product of two factors,
one of which should be the largest perfect square.

1) Simplify the following surds. You MUST show full working out. DO NOT use your calculator.

a) √98 b) √88 c) √128

2) Express 2√80 in the form 𝑐√5 where c is a constant.

3) Express 3√32 in the form 𝑎√𝑏 where a and b are integers.

4) Write 7√20 in the form 𝑘√5, where k is an integer.

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1.5 ADDITION AND SUBTRACTION OF SURDS
You can only add or subtract ‘like’ surds.
This implies that, to add or subtract surds, you need to simplify them first.

1. √8 + √18

2. √75 − √12

3. √108 + √75

4. √50 + √18 − √72

5. √32 + √8 − √2

6. √45 + √500 − √80

7. Express √48 + √108 in the form 𝑘√3.

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1.6 EXPANDING BRACKETS CONTAINING SURDS
Simplify the following by removing the brackets:

1. √3(4√3 + 2) 2. (√5 + 2)(√5 − 2)

2
3. (√6 + 1) 4. (2√3 + 1)(√3 − 1)

2
5. √7(3√7 − 2) 6. (√7 − √2)

7. (√7 − √2)(√7 + √2) 8. (9 − √2)(9 + √2)

2 2
9. (√3 + 1) + (√3 − 1)

10. Calculate the volume of the cuboid shown. Give your answer as a surd in its simplest form.

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 2
8. (𝑎 + √𝑏) = 49 + 12√𝑏 where 𝑎 and 𝑏 are integers, and 𝑏 is prime. Find the value of 𝑎
and the value of 𝑏.

9. (3 + √𝑎)(4 + √𝑎) = 17 + 𝑘√𝑎 where 𝑎 and 𝑘 are positive integers. Find the value of 𝑎
and the value of 𝑘.

2
10. (√𝑎 + √8𝑎) = 54 + 𝑏√2 where 𝑎 and 𝑏 are positive integers. Find the value of 𝑎 and the
value of 𝑏.

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1.7 RATIONALIZING THE DENOMINATOR
This means eliminating all surds in the denominator.
1. Write as a surd in its simplest form with a rational denominator
Hint: Multiply the numerator and the denominator by the surd in the denominator.

8 15
a) b)
√20 2√5

√2 5√7
c) d)
√5 4√3

5+√7 2√3
e) f)
3 √7 √7

7+√3 7−√3
g) h)
2√7 3√11

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 5−√10 −5−√10
i) j)
2√10 −2√7

Given an expression √𝑎 + √𝑏, the conjugate of this expression is √𝑎 − √𝑏.

Notice that the conjugate has the same terms but opposite sign in the middle.
2. Write as a surd in its simplest form with a rational denominator
Hint: Multiply the numerator and the denominator by the denominator’s conjugate.

3
a)
3+√5

√7
b)
2−√3

5 √5
c)
5+√11

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 6+√7
d)
5−√3

3 √3
e)
√7+14

2+√3
f)
3+√5

√10
g)
3−√2

5+√5
h)
5−√3

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 3−√7
i)
5−√2

3. A rectangle has an area of 50 cm2. If the length of the rectangle is √40, find the width giving
your answer as a surd with a rational denominator.

4. If 𝑥 = √2 + 1 and 𝑥 2 𝑦 = 1, find 𝑦 as a surd in its simplest form with a rational

denominator.

5. A cuboid has a square base of side √3 + 2. The volume of the cuboid is 60 cm2. Find the
height as a surd in its simplest form with a rational denominator.

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 4
6. In a right-angled triangle, sin 𝐴 = 6, find the exact values of cos 𝐴 and tan 𝐴.

1
7. In a right-angled triangle, sin 𝑥 = 2. Write cos 𝑥 and tan 𝑥 as surds in their simplest form.

2
8. If cos 𝑥 = 3, find sin 𝑥 and tan 𝑥 in their simplest surd form.

3
9. If tan 𝑥 = 5, find cos 𝑥 and sin 𝑥 in their simplest surd form.

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10. Find 𝑥 in its simplest form with a rational denominator.

11. Find 𝑥 in its simplest form with a rational denominator.

𝑎+√4𝑏
12. Rationalize the denominator of where 𝑎 is an integer and 𝑏 is a prime number.
𝑎−√4𝑏
Simplify your answer.

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