Addition and Subtraction of Radical Expressions

 

Arithmetic of Radical Expressions

When we add, subtract, and multiply expressions that involve radicals we do so just as we do arithmetic on polynomials.  we use the distributive law, combine like terms and FOIL.

Example:  

 

  1. 3( - ) + 4( +

    = 3 - 3 + 4 + 4

    = 7 +

  2. ( - )2  

    = ()2 - 2 + ()2

    = 3 - 2 + 5 = 8 - 2

     

 

Exercises:  

Perform the indicated operations and simplify

 

  1. ( - )( + )    -4

  2. 3( + 5 )2               456 + 60 sqrt(3)


Rationalizing The Denominator


When we divide with radicals in the denominator, we multiply the numerator and denominator by the conjugate root

 

Definition

If

          

is and expression, then its conjugate is 

         

 

Example

        +                   ( + )( + )
                                =                                                                                   
        -                     ( - )( + )

        =    2 + 2 + 3                     5 + 2 
                                            =                                                                       
                     2 - 3                                 -1

        =     -5 - 2

 


Exercises  

Rationalize the denominator

 

  1.     2 +
                           
    (-3 + 2sqrt(5) - 2sqrt(3) + sqrt(15))/2
     
    +        

  2.   3 - 4
                             (9sqrt(2) + 2sqrt(15))/4
     2 - 4
                               

  3.       15                    3 
                        +                     
    Hint:  Common Denominators
     4 +
                    
            
                    (132 - 30sqrt(11))/11


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