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What Is A Surd?

What Are Surds?

Rational
Numbers
Can be written in the form
a

b
where a and b are integers
eg
1

2
and
3

1

Irrational
Numbers
Can NOT be written in the form
a

b
where a and b are integers
 eg √2 and pi

Surds Irrational numbers such as  √2, √3, √5, √6

 

Simplify Surds

√(ab) = √a√b

NUMERIC EXAMPLE
√(16 × 9) = √16√9= 4 × 3 = 12
√(16 × 9) = √(144) = 12

 

Using √(ab) = √a√b

EXAMPLE
√12 = √(4 × 3) = √4√3= 2√3

 

LEVEL 1

FIRST
Find the square number
SECOND
Express the surd as a product of the square number and another number
THERE IS LESS CHANCE OF ERROR IF YOU WRITE THE SQUARE NUMBER
BEFORE THE OTHER NUMBER

Example 1
√12      square number is 4
= √(4 × 3)
= √4√3      USE THIS STEP TO AVOID ERROR= 2√3

Example 2
√50      square number is 25
= √(25 × 2)
= √25√2
= 5√2

Example 3
√80      square number is 16
= √(16 × 5)
= √16√5
= 4√5

 

LEVEL 2

FIRST
Find the square number
SECOND
Express the surd as a product of the square number and another number
THERE IS LESS CHANCE OF ERROR IF YOU WRITE THE SQUARE NUMBER
BEFORE THE OTHER NUMBER

Example 1
3√20      square number is 4
= 3√(4 × 5)
= 3√4√5
= 3 × 2√5    USE MULTIPLICATION SIGN TO AVOID ERROR
= 6√5

Example 2
2√300      square number is 100
= 2√(100 × 3)
= 2√100√3
= 2 × 10√3
= 20√3

Example 3
5√72      square number is 36
= 5√(36 × 2)
= 5√36√2
= 5 × 6√2
= 30√2

 

LEVEL 3

FIRST
Find the square number for each surd
SECOND
Express each surd as a product of the square number and another number
THERE IS LESS CHANCE OF ERROR IF YOU WRITE THE SQUARE NUMBER
BEFORE THE OTHER NUMBER

Example 1
√72 + √12      square numbers are 36 and 4
= √(36 × 2) + √(4 × 3)
= √36√2 + √4√3
= 6√2 + 2√3

Example 2
√40 - √32      square numbers are 4 and 16
= √(4 × 10) - √(16 × 2)
= √4√10 - √16√2
= 2√10 - 4√2

Example 3
- √8 + √20      square numbers are 4 and 4
= - √(4 × 2) + √(4 × 5)
= - √4√2 + √4√5
= - 2√2 + 2√5
= 2√5 - 2√2

 

LEVEL 4

FIRST
Find the square number for each surd
SECOND
Express each surd as a product of the square number and another number
THERE IS LESS CHANCE OF ERROR IF YOU WRITE THE SQUARE NUMBER
BEFORE THE OTHER NUMBER

Example 1
2√72 + √32      square numbers are 36 and 16
= 2√(36 × 2) + √(16 × 2)
= 2√36√2 + √16√2
= 2 × 6√2 + 4√2
= 12√2 + 4√2
= 16√2

Example 2
√40 - 5√32      square numbers are 4 and 16
= √(4 × 10) - 5√(16 × 2)
= √4√10 - 5√16√2
= 2√10 - 5 × 4√2
= 2√10 - 20√2

Example 3
- 14√8 + 7√20      square numbers are 4 and 4
= - 14√(4 × 2) + 7√(4 × 5)
= - 14√4√2 + 7√4√5
= - 14 × 2√2 + 7 × 2√5
= - 28√2 + 14√5
= 14√5 - 28√2

 

LEVEL 5

FIRST
Find the square number for each surd
SECOND
Express each surd as a product of the square number and another number
THERE IS LESS CHANCE OF ERROR IF YOU WRITE THE SQUARE NUMBER
BEFORE THE OTHER NUMBER

Example 1
√72 + √48 + √27       square numbers are 36, 16 and 9
= √(36 × 2) + √(16 × 3) + √(9 × 3)
= √36√2 + √16√3 + √9√3
= 6√2 + 4√3 + 3√3
= 6√2 + 7√3

Example 2
2√72 + 3√48 + 4√27       square numbers are 36, 16 and 9
= 2√(36 × 2) + 3√(16 × 3) + 4√(9 × 3)
= 2√36√2 + 3√16√3 + 4√9√3
= 2 × 6√2 + 3 × 4√3 + 4 × 3√3
= 12√2 + 12√3 + 12√3
= 12√2 + 24√3

Example 3
- 2√72 + 3√48 - 4√27       square numbers are 36, 16 and 9
= - 2√(36 × 2) + 3√(16 × 3) - 4√(9 × 3)
= - 2√36√2 + 3√16√3 - 4√9√3
= - 2 × 6√2 + 3 × 4√3 - 4 × 3√3
= -12√2 + 12√3 - 12√3
= -12√2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Edited by Dr David Cornelius an Independent Private Maths Tutor with over 25 years of experience and The Secretary of The Association of Tutors in the UK for 15 years.
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