surds

51

34

-23 0.59

Rational Numbers Irrational Numbers

Any number which can be written as a fraction

Any number which cannot be written as a fraction

π = 3.1415926....

SurdsWhat is a surd?

To answer this question we must first look at rational and irrational numbers

Note

√8 is a surd but ∛8 is not since ∛8 = 2

Lets now turn our attention to square rooting numbers

SurdsWhat is a surd?

Square Numbers Everything else

√16 = 4√9 = 3√4 = 2√1 = 1

√25 = 5√81 = 9√64 = 8√49 = 7√36 = 6

√100 = 10

√2 = 1.41421....

√5 = 2.23606....

√32 = 5.65685....

√97 = 9.84885....

Answers are rational Answers are irrational

A surd is any root that gives an irrational answer

√81

x2 = 42 + 52

x2 = 41x = √41

SurdsExamples

Which of the following are surds

Express x as a surds in each case

x4

5

x8

5x2 = 82 - 52

x2 = 39x = √39

Solve each equation giving answers in surd form

x2 + 4 = 6 x2 - 2 = 9 3x2 - 5 = 10x2 = 2x = √2

x2 = 11x = √11

3x2 = 15x2 = 5x = √5

Write the exact values of each ratio in surd form

x√2

1

x

√3

5tanx = √2 sinx = √3 5

√13 √144 √3 ∛125 √9

Think 2x + 4x = 6xThink x + x = 2x

Surds can be simplified using the normal rules of algebra

SurdsSimplifying surds

Examples

√3 + √3 = 2√3 2√5 + 4√5 = 6√5

Add or subtract these surds

6√2 + 3√2 = 9√2 4√2 - 3√2 = √2

9√7 + 12√7 = 21√7 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

√4 x √9 = √362 3 6


√a x √b = √abGenerally

Examples

√40 = √2 x √20 √40 = √4 x √10or

√18 = √3 x √6 or √18 = √9 x √2

√72 = √9 x √8 or √72 = √36 x √2

√50 = √5 x √10 or √50 = √25 x √2

There are different factors to choose from but we always look for square number factors to simplify surds

Biggest square factor

Surds

Examples

√20 = √4 x √5

Simplify the surds below

= 2 x √5√48 = √16 x √3

= 4 x √3 = 2√5 = 4√3

Memorise your square numbers!

1 4 9 16 2536 49 64 81 100

Biggest square factor

Find the exact value of y as a surd in its simplest form

y

6cm

4cm

y2 = 42 + 62

y2 = 52y = √52y = √4 x √13y = 2√13

Simplify


√2 x √2 = 2!!!Remember

Examples

√3 x √6 = √18

Simplify as far as possible

√18 = √9 x √2= 3√2

Find a square factor

√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

√5

7

x2 = 72 + (√5)2

x2 = 49 + 5x2 = 54x = √54x = √9 x √6x = 3√6

Simplify


1 4 9 16 2536 49 64 81 100

√a x √b = √ab

SurdsHarder Examples

Examples

(2 +√2)(3 +√2) Multiply out and simplify


1 4 9 16 2536 49 64 81 100

= 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

SurdsMultiplying by 1


1 4 9 16 2536 49 64 81 100

45 x 1 = 4

5Of course this is true!!!!!!

45 x 2

2 = 810 = 4

5Again I have multiplied by 1

45 x a

a = 4a5a = 4

5Again I have multiplied by 1

It is good practice in Maths to write fractions with rational denominator

SurdsRationalising the denominator

Examples


1 4 9 16 2536 49 64 81 100

51

34

-23

√23

5√32 Rational numbers

What do we do if the denominator is irrational?

Rationalise the denominator in each fraction

5√3 √3

√3x3

5√3=

3√2 √2

√2x2

3√2=

Rationalise the denominator and simplify as far as possible

8√2 √2

√2x2

8√2= = 4√2

103√5 √5

√5x3 x 510√5

= =15

10√53

2√5=

= 1

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