Surds
Objectives:
Grade A:
Rationalise the denominator of a surd, such as
Grade A*:
Simplify surds such as write
in the form
Surds
Prior knowledge:
1.
2.
3.
4.
5.
6.
7.
8.
52
152
132
112
92
√144
√196
√49
Identify factors that are square numbers:
25
225
169
121
81
12
14
7
1.
2.
3.
4.
5.
6.
7.
8.
20
32
500
45
72
18
98
80
4×5
16 × 2
100 × 5
9×5
36 × 2
9×2
49 × 2
16 × 5
Surds
Rational number
a rational number is a number which can be expressed
as a ratio of two integers. Such that
Where a and b are integers
e.g.
=
Irrational number
an irrational number is a number that is not an integer and
cannot be expressed as a fraction.
e.g.
,
,
,π
Surd
a surd is a number containing an irrational root,
e.g.
or 3 + 2
Surds
Why use surds?
To write an irrational number in surd form is to write it
accurately.
e.g. π is an irrational number, it has an infinite number of decimal places
if you were to write it on paper as digits, at some point you would have
to round it and then it would stop being completely accurate.
We can find out whether a number is rational or irrational by
simplifying it as far as possible.
If we have simplified as far as possible and we are left with a
number such as a square root of a number that is not a natural
square number or a value such as π then we can determine that
the number is irrational
Surds
Example:
Are these numbers rational or irrational?
√5
irrational – because it cannot be simplified to an integer
√ 36
π
2
π
2π
rational – because it can be simplified to an integer √ 36 = 6
irrational – because π is irrational
rational – because it can be simplified to 1
2
Surds
When simplifying surds it is useful to know the following
√4 = 2
2×2=4
therefore
and
Also because 4 = √ 16
√4 × √4 = 4
√2 × √2 = 2
√ 4 × √ 4 = √ 16
Rule 1
√ a × √ b = √ ab
Example:
Simplify √ 98
Firstly we will look for factors of 98
that are square numbers 49 × 2 = 98
so √ 98 can be written as √ 49 × 2
Using the rule
√ 49 × √ 2
=7√ 2
Surds
Simplify these:
1.
√ 20
√4 × 5
=√ 4 ×√ 5 = 2√ 5
2. √ 32
√ 16 × 2 =√ 16 ×√ 2 = 4√ 2
3. √500
√ 100 × 5 = √ 100 ×√ 5 = 10√ 5
4.
√ 45
√9 × 5
5.
√ 72
√ 36 × 2 =√ 36 ×√ 2 = 6√ 2
6.
√ 18
√9 × 2
=√ 9 × √ 5 = 3√ 5
=√ 9 × √ 2 = 3√ 2
Surds
Rule 2
a
√b
Example:
Simplify
√a
=
√b
96 √ 96
=
√3
√ 3
using rule 2
Find factor(s) of 96 that are square numbers
=
6 = 2 × 3 so
√16×6
√3
√6 = √2 × √3
so 4 ×√ 6
√3
4 ×√ 6
√16 ×√ 6
=
=
√3
√3
Rule √ a × √ b = √ ab
4 ×√ 2×√ 3
=
√3
=4√ 2
Surds
Simplify these:
1.
40
√ 5
=√ 8
2.
81
√ 16
√ 81
=
√ 16
3.
45
√ 20
9
√4
√9
=
√4
4.
72
√ 16
9
√2
=
√ 4 × 2 =√ 4 ×√ 2 = 2√ 2
= 9 = 21
4
4
√9
√2
= 3 = 11
2
2
=3
√2
Surds
Simplify these:
5.
6 √10 = 3 ×√ 10 =3 ×√ 2×√ 5
2√5
√5
√5
6.
8 √30
4 √5
7.
√ 7×√ 3
2 √21
=
4 √3
8 √3
8.
2 √10
10√ 2
= 3√ 2
2 ×√ 30 =2 ×√ 6×√ 5 = 2 ×√ 2×√ 3
=
√5
√5
=
√10
5 √2
√7
=
4
√ 2×√ 5
=
5 √2
√5
=
5
= 2√ 2 √ 3
Surds
Rule 3
a √ c + b√ c = (a + b)√ c
Example:
Simplify
3 √ 2 + √ 32
= 3 √ 2 + √ 16 × 2
= 3 √ 2 + √ 16 ×√ 2
= 3√2 + 4 √2
= 7√ 2
Surds
Simplify these:
1.
3√2 + √2
= 4√ 2
2.
3 √ 5 - 2√ 5
= √5
3.
√ 20 +√ 5
4.
3 √ 7 - √ 28 = 3√ 7 - √ 4 × 7 = 3√ 7 - √ 4 ×√ 7
= √ 4 × 5 + √ 5 = √ 4 ×√ 5 + √ 5
= 2 √ 5 + √ 5 = 3√ 5
= 3√ 7 - 2 √ 7
= √7