Surds
What is a surd?
When the square root of a number is a decimal number (and not a whole number), it is called a surd.
Examples of surds are:
- 2 = 1.4
- 3 = 1.7
- 5 = 2.2
- 6 = 2.4
- 7 = 2.6
What is NOT a surd?
When a square root of a number is a whole number, it is not a surd.
Examples that are not surds are:
- 1 = 1
- 4 = 2
- 9 = 3
Maths Fun - Compass Activity
- Draw a right-angled triangle. Make each of the sides that form the right angle 1 unit long.
- Using Pythagoras's Rule, the hypotenuse is 2.
- Extend a compass to the length of 2 and draw the next right-angled triangle with a side of 1 and the second side 2.
- Using Pythagoras's Rule, the hypotenuse of the next triangle is 3.
- Continue this to find the square roots of whole numbers.
- Consider that the ancient Greeks probably used this method to work square roots of surds and non-surds. Clever, isn't it?
Questions - Which are Surds?
Find the square roots of these numbers - 1, 2, 3, 4, 5, 25, 50, 100.
Which are surds?
Answer
Surds are the square roots of 2, 3, 5, 50.
Example One - Simplify Surds
Simplify:
(a) 8
(b) 27
(c) 48
(d) 50
Answers:
Firstly, find the factors of the number where one of the factors is a square number such as 4, 9, 16, 25 and so on. Then simplify.
(a) 8 = 4 × 2 = 22
(b) 27 = 9 × 3 = 33
(c) 48 = 16 × 3 = 43
(d) 50 = 25 × 2 = 52
Questions - Simplify these Surds
(a) 12
(b) 45
(c) 500
(d) 75
Answers
(a) 23
(b) 35
(c) 105
(d) 53
Example Two - Entire Surds
Make these entire surds:
(a) 32
(b) 25
(c) 35
(d) 57
Answers:
This is the reverse of Example One above.
(a) 32 = 9 × 2 = 18
(b) 25 = 4 × 5 = 20
(c) 35 = 9 × 5 = 45
(d) 57 = 25 × 7 = 175
Questions - Make Entire Surds
(a) 53
(b) 210
(c) 37
(d) 73
Answers
(a) 75
(b) 40
(c) 63
(d) 147
Example Three - Simplify these Multiplied Surds
Simplify these surds:
(a) 23 × 53
(b) 25 × 52
Answers:
(a) 23 × 53
= 4 × 3 × 25 × 3
= 900
= 30
(b) 25 × 52
= 4 × 5 × 25 × 2
= 1000
= 100 × 10
= 1010
Example Four - Simplify these Divided Surds
Simplify these surds:
| (a) | 85 |
| 25 |
| (b) | 67 × 53 |
| 23 |
Answers:
| (a) | 85 | = | 8 | = 4 |
| 25 | 2 |
| (b) | 67 × 53 | = | 67 × 5 | = | 37 × 5 | = | 157 |
| 23 | 2 |
Questions - Simplify these Surds
(a) 23 × 53
| (b) | 43 |
| 26 |
Answers
(a) 30
(b) 2
Example Five - Simplifying Surds on the Bottom of a Fraction
Simplify these surds:
| (a) | 3 |
| 2 |
| (b) | 5 |
| 3 |
Answers:
If there is a surd on the bottom of a fraction, multiply both the top and the bottom of the fraction by that surd. Then simplify.
| (a) | 3 | = | 32 | = | 32 |
| 2 | 2 × 2 | 2 |
| (b) | 5 | = | 53 | = | 53 |
| 3 | 3 × 3 | 3 |
Questions - Simplify these Surds
| (a) | 7 |
| 2 |
| (b) | 8 |
| 2 |
Answers
| (a) | 7 2 |
| 2 |
(b) 42
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