PURE MATHEMATICS - Algebra

 

Surds

 

 

useful expressions rationalising reduction diff. two squares rational, irrational

 

 

Rules

 

Surds are mathematical expressions containing square roots.

 

However, it must be emphasized that the square roots are 'irrational'

 

i.e. they do not result in a whole number, a terminating decimal or a recurring decimal.

 

 

The rules governing surds are taken from the Laws of Indices.

 

Rule #1

               

 

Examples

 

 

Rule #2

                       

 

Examples

 

back to top

 

 

Some Useful Expressions

 

 

Expression #1

                     

             

 

                       

             

 

example

simplify        

 

                      

           

              

       

 

ans.    

 

back to top

 

 

Expression #2 - (the difference of two squares)

 

 

         

 

 

 

 

 

example

evaluate    

 

 

                         

                   

 

ans.

 

 

back to top

 

 

Rationalising Surds

 

This is a way of modifying surd expressions so that the square root is in the numerator of a fraction and not in the denominator.

 

The method is to multiply the top and bottom of the fraction by the square root.

 

 

 

 

 

back to top

 

 

Rationalising expressions using 'Difference of Two Squares'

 

 

Remembering that :

 

.....from 'useful expressions' above.

 

 

Example #1 -                                     simplify

 

 

Multiplying top and bottom by  

 

 

 

 

 

 

 

 

back to top

 

Example #2 - rationalise               

 

 

Multiply top and bottom by   

 

                              

                              

 

                                             

 

 

                    

 

 

 

 

back to top

 

 

Reduction of Surds

 

This is a way of making the square root smaller by examining its squared factors and removing them.

 

 

         

 

                            

 

back to top

 

 

Rational and Irrational Numbers

 

Test for rational and irrational numbers:

 

If a surd has a square root in the numerator, while the denominator is '1' or some other

 

number, then the number represented by the expression is 'irrational'.

 

 

examples of irrational surds:

 

                            

 

 

 

back to top

 

 

creative commons license

All downloads are covered by a Creative Commons License.
These are free to download and to share with others provided credit is shown.
Files cannot be altered in any way.
Under no circumstances is content to be used for commercial gain.

 

 

 

 

©copyright a-levelmathstutor.com 2020 - All Rights Reserved