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:


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:

Maths Fun - Compass Activity

Square Roots with Compass

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
252
(b)67 × 53 = 67 × 5 = 37 × 5 = 157
232

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
22 × 22
(b)5 = 53 = 53
33 × 33

Questions - Simplify these Surds

(a)7
2
(b)8
2

Answers

(a)7 2
2

(b) 42