IL1.
5: SURDS
Definition
A surd is an irrational number resulting from a radical expression that cannot be evaluated directly.
1
For example 2, 3, , 20 , 3
2, 3
9, 4
8 , etc are all surds
8
1
but 1, 9, , 3
8, 4
16 , 3
64 are not.
100
a
Surds cannot be expressed exactly in the form , and can only be approximated by a decimal.
b
eg: 2 ≈ 1.414
See Exercise 1
Simplifying Surds
If a surd has a square factor ie 4, 9, 16, 25, 36…, then it can be simplified.
Examples
1. 200 = 100 × 2
= 100 × 2 [NB: It is best to find the largest square number factor!]
= 10 2
2. 3 48 = 3 16 × 3
= 3 16 × 3
= 3×4× 3
= 12 3
3
24 8× 3
3
3. =
4 4
3
8× 3 3
=
4
2× 3 3
=
4
3
3
=
2
See Exercise 2
IL1.5 - Surds Page 1 of 6 June 2012
Addition and Subtraction
Only like surds can be added or subtracted
Examples
1. 8 5 -3 5 =5 5
2. 2 3 + 5 7 - 10 3 = 5 7 - 8 3
Sometimes it is necessary to simplify each surd first
3. 18 - 8 − 20 = 9× 2 − 4× 2 − 4×5
= 3 2 −2 2 −2 5
= 2 −2 5
NB: 5 + 7 ≠ 12 because 5 and 7 are not ‘like’ surds.
See Exercise 3
Multiplication
To multiply surds the whole numbers are multiplied together, as are the numbers enclosed by the radical sign
ie
a b ×c d =
ac bd
Examples:
1. 2× 3 =6
2. 3 3×4 5 =
12 15
Sometimes it is possible to simplify after multiplication
3. 2 10 × 7 6 =
14 60
= 14 4 × 15
= 14 × 2 15
= 28 15
See Exercise 4
IL1.5 - Surds Page 2 of 6 June 2012
Expansion of Brackets
The usual algebraic rules for expansion of brackets apply to brackets containing surds
a(b + c) = ab + ac
and
(a + b)(c + d) = ac + bc +ad + bd
Examples
1. 2 ( )
2 + 5 =2 + 5 2
2. 2 3 ( )
3 − 3 2 =2 9 − 6 6
=6-6 6
3. (7- 5) 2 =
(7 − 5)(7 − 5)
= 49 − 7 5 − 7 5 + 5
= 54 − 14 5
4. ( )(
6 −4 3 2 2 −3 5 = ) 2 12 − 8 6 − 3 30 + 12 15
= 2 4 × 3 − 8 6 − 3 30 + 12 15
= 2 × 2 × 3 −8 6 − 3 30 + 12 15
= 4 3 −8 6 − 3 30 + 12 15
See Exercise 5
Fractions containing surds
Surds expressed as fractions may be simplified using
a a
=
b b
Examples
18 18
1. = = 3
6 6
4 16 16
2. = = = 8 or 2 2
2 2 2
See Exercise 6
IL1.5 - Surds Page 3 of 6 June 2012
Rationalizing surds
Rationalized surds are expressed with a rational denominator.
Examples
2 2 5
1. = ×
5 5 5
2 5
=
5
5 5 2
2. = ×
3 2 3 2 2
10
=
6
Conjugate surds
The pair of expressions a + b and a − b are called conjugate surds. Each is the conjugate of the
other.
The product of two conjugate surds does NOT contain any surd term!
( a + b) ( a - b) = a–b
( )( ) ( 10 ) − ( 3 )
2 2
Eg: 10 − 3 10 + 3 =
= 10 – 3
= 7
We make use of this property of conjugates to rationalize denominators of the form
a + b and a− b.
Example
3 3 5− 2
= ×
5+ 2 5+ 2 5− 2
=
(
3 5− 2 )
25 − 2
5 3− 6
=
23
See Exercise 7
IL1.5 - Surds Page 4 of 6 June 2012
Operations on fractions that contain surds
When adding and subtracting fractions containing surds it is generally advisable to first rationalize each fraction:
Example
2 1 2 3 2 −1 1 3+ 2
+ = × + ×
3 2 +1 3− 2 3 2 +1 3 2 −1 3− 2 3+ 2
= 6 2 −2 3+ 2
+
18 − 1 3− 2
= 6 2 −2 3+ 2
+
17 1
= 6 2 − 2 17( 3 + 2)
+
17 17
= 6 2 − 2 + 17 3 + 17 2
17
= 23 2 − 2 + 17 3
17
See Exercise 8
Exercises
Exercise 1
Decide whether the following radical expressions are rational or irrational and evaluate exactly if rational:
1
(a) 25 (b) 2.25 (c) (d) 3
64 (e) 12 ×18
10
Exercise 2
Simplify
2
(a) 20 (b) 48 (c) 500 (d) 2 12 (e)
20
Exercise 3
Simplify
(a) 2 3 +5 3 (b) 3 7 +5 7 −3 7
(c) 2 + 7 +3 2 −4 7 (d) 54 − 24
Exercise 4
Simplify
(a) 3 × 10 (b) 2 7 ×5 3
(c) 10 × 3 10 (d) 2 8 × 2 50 × 2
Exercise 5
Expand the brackets and simplify if possible
(a) 2 ( 2 −8 ) (b) ( 2 + 3 )( 3 − 4)
(1 + ) ( 11 + 3)( 11 − 3)
2
(c) 10 (d)
IL1.5 - Surds Page 5 of 6 June 2012
Exercise 6
Simplify
12 32
(a) (b)
6 2
3 28 4 2 ×3 3
(c) (d)
7 6 6
Exercise 7
Express the following fractions with a rational denominator in simplest form:
5 1 2 18
1. (a) (b) (c)
2 10 8
2+ 3 1 3+2
2. (a) (b) (c)
2 3− 2 3−2
Exercise 8
2 3
Evaluate and express with a rational denominator +
3 −1 2 − 3
Answers
Exercise 1
(a) 5 rational (b) 1.5 rational (c) irrational (d) 4 rational (e) irrational
Exercise 2
1
(a) 2 5 (b) 43 (c) 10 5 (d) 4 3 (e)
5
Exercise 3
(a) 7 3 (b) 5 7 (c) 4 2 - 3 7 (d) 6
Exercise 4
(a) 30 (b) 10 21 (c) 30 (d) 80 2
Exercise 5
(a) 2-8 2 (b) -5-2 3 (c) 11+2 10 (d) 2
Exercise 6
(a) 2 (b) 4 (c) 6 (d) 2
Exercise 7
10 10
1. (a) (b) (c) 3
2 10
2 2+ 6
2. (a) (b) 3+ 2 (c) −7 − 4 3
2
Exercise 8
4 3+7
IL1.5 - Surds Page 6 of 6 June 2012
