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Number (Higher) - Surds
 
Definition | Simplifying Surds | Changing the Form of a Surd | Problems | Rationalising the Denominator
 
Definition

When the square root of a number is the only exact way of writing down the number, it is called a surd.

So square_root_02 is called a surd.

 
Simplifying Surds

a. Adding and subtracting

Rule: We can add or subtract the square roots of the same numbers only.

square_root_02+square_root_02 
   = 2 square_root_02
     

5 square_root_02 – 2 square_root_02 
   = 3 square_root_02
     

square_root_02+squareroot_3large 
   = cannot be simplified.


b. Multiplying and Dividing

Rule: We can multiply and divide surds.
 

square_root_02x squareroot_3large 
   = squareroot_6large
 

squareroot_6large÷ squareroot_3large 
  
=  square_root_02
       

Note:

 squareroot_6large÷ 2
   cannot be simplified.
But

squareroot_6largex 2
   = 2 squareroot_6large

We must be aware of this when multiplying or dividing surds by integers.

 
Changing the Form of a Surd
   
squareroot_24large= squareroot_4x6large= square root 4 proper x squareroot_6large= 2 x squareroot_6large = 2 squareroot_6large
So
  
 squareroot_24large= 2 squareroot_6large

If one of the factors of the number inside the square root is a square number, then it can be square rooted and taken outside.
 
 squareroot_27large= squareroot_9x3large= 3 squareroot_3large

 
Problems on Surds

Example 1: A rectangle has a length of (2 + squareroot_3large)cm and width (3 – squareroot_3large)cm. Work out the area of the rectangle.

  Area = (2 + squareroot_3large) (3 – squareroot_3large) = 6 – 2 squareroot_3large +3 squareroot_3large – 3
  Area = (3 + 3 squareroot_3large) cm

Example 2: Simplify squareroot_8 x squareroot_3large =
= squareroot_8 x squareroot_3large= squareroot_24large= 2 squareroot_6large

 
Rationalising the Denominator

This is a way of removing the surd from the denominator.
Example 1:
3
=
3
x
  square_root_02 = 3 square_root_02  
 

square_root_02

  
square_root_02
   _____
square_root_02		  
2
  

Rule: To rationalise the denominator, multiply above and below by the surd.

Example 2: When the denominator is an expression.

 
  5  
=
    5   
x
 ( 2 – squareroot_3large) = 5 ( 2 – squareroot_3large)
 
(2 + squareroot_3large)
  
( 2 + squareroot_3large)
   ( 2 – squareroot_3large)   4 – 2   squareroot_3large+ 2 squareroot_3large – 3
               
   
=
 5(2 – squareroot_3large)
=
 5 ( 2 – squareroot_3large)    
     
1  
        

Rule: Multiply above and below by the expression with the sign of the surd changed. This eliminates the surd from the denominator.