Surds What is a surd?
To answer this question we must first look at rational and irrational numbers
Rational Numbers 5 1
3 -2 3 4
Irrational Numbers
0.5
Ď€ = 3.1415926....
Any number which can be written as a fraction
Any number which cannot be written as a fraction
9
Surds What is a surd? Lets now turn our attention to square rooting numbers
Square Numbers √1 = 1 √4 = 2 √9 = 3 √16 = 4 √25 = 5
Everything else
√36 = 6 √49 = 7 √64 = 8 √81 = 9 √100 = 10
√2 = 1.41421.... √5 = 2.23606.... √32 = 5.65685....
√97 = 9.84885.... Answers are irrational
Answers are rational
A surd is any root that gives an irrational answer Note
√8 is a surd but ∛8 is not since ∛8 = 2
Surds Examples Which of the following are surds
√81
√144
√13
Express x as a surds in each case
4
2
2
∛125
√3
2
5
x2 = 82 - 52
x = 4 + 5
x
x
x2 = 41 x = √41
5
√9
x2 = 39 x = √39
8
Solve each equation giving answers in surd form
x2 + 4 = 6
x2 = 2 x = √2
x2 - 2 = 9
3x2 - 5 3x2 x2 x
x2 = 11 x = √11
= = = =
10 15 5 √5
Write the exact values of each ratio in surd form
√2 x
1
5
tanx = √2
x
√3
3 sinx = √ 5
Surds Simplifying surds Surds can be simplified using the normal rules of algebra
√3 + √3 = 2√3
2√5 + 4√ 5 = 6 √ 5
Think x + x = 2x
Think 2x + 4x = 6x
Examples Add or subtract these surds
6√2 + 3√2 = 9√2 9√7 + 12√7 = 21√7
4√2 - 3√2 = √2
7√3 - 9√3 = -2√3
3√5 + 5√5 + 2√5 + 4√5 = 14√5 8√2 - 6√2 + 9√2 - 7√2 = 4√2
Surds Simplifying surds
√4 x √9 = √36 2
6
3
Generally
√a x √b = √ab Examples
√40 = √2 x √20
or
√40 = √4 x √10
√18 = √3 x √6
or
√18 = √9 x √2
√50 = √5 x √10
or
√50 = √25 x √2
√72 = √9 x √8
or
√72 = √36 x √2
There are different factors to choose from but we always look for square number factors to simplify surds
Surds Memorise your square numbers!
1 4 9 16 25 36 49 64 81 100 Examples Simplify the surds below Biggest square factor
Biggest square factor
√20 = √4 x √5
√48 = √16 x √3
= 2 x √5
= 4 x √3
= 2√ 5
= 4 √3
Find the exact value of y as a surd in its simplest form
y2 = 42 + 62
4cm
y
y2 = 52 y = √52
6cm
Simplify
y = √4 x √13 y = 2√13
Surds
Memorise your square numbers!
1 4 9 16 25 36 49 64 81 100
Simplifying surds
Remember
√ !!! √b2 ==√2ab √a2 xx √ Examples
Simplify as far as possible
√3 x √6 = √18 √18 = √9 x √2
Find a square factor
= 3√ 2
√2 x √5 x √6 = √60
√60 = √4 x √15
= 2√15 Find the exact value of x as a surd in its simplest form
x 2 = 7 2 + ( √5 ) 2
x
7 √5
x2 = 49 + 5 x2 = 54 x = √54 Simplify x = √9 x √6 x = 3√ 6
Surds
Memorise your square numbers!
1 4 9 16 25 36 49 64 81 100
Harder Examples
Examples
Multiply out and simplify
(2 +√2)(3 +√2) = 6 + 2 √2 + 3 √2 + 2
= 8 + 5√ 2
Multiply out and simplify
(1 +√3)(4 -√3) = 4 - √3 + 4 √3 - 3
= 1 + 3√ 3
Surds
Memorise your square numbers!
1 4 9 16 25 36 49 64 81 100
Multiplying by 1
4 4 x 1 = 5 5 Of course this is true!!!!!!
2 4 8 4 = = x 2 5 5 10 Again I have multiplied by 1
a 4 4 4 a = = x a 5 5 5a Again I have multiplied by 1
Surds
Memorise your square numbers!
1 4 9 16 25 36 49 64 81 100
Rationalising the denominator
It is good practice in Maths to write fractions with rational denominator
5 1
3 -2 √2 5√3 4 3 3 2
Rational numbers
What do we do if the denominator is irrational?
Examples Rationalise the denominator in each fraction
5 3 5 3 x √ = √ 3 √3 √3
3 2 3 2 x √ = √ 2 √2 √2
= 1
Rationalise the denominator and simplify as far as possible
8 2 8 2 x √ = √ 2 √2 √2
= 4√2
10 5 10√5 x √ = 3√5 3 x 5 √5
=
10√5 2√5 = 15 3