Surds

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