Question

solution for this question

Answer 1

                                            ☺☺☺

Sol :     1                    2                           1
         ----------  +    --------------    +    --------------  = 0.
        ( 2 + √3)      ( √5 - √3 )             ( 2 - √5 )

We have, 1 / ( 2 + √3 ).

To rationalize the denominator , we have to multiply the numerator and denominator by ( 2 - √3 ).

So, 1  ( 2 - √3 ) / ( 2 + √3 ) ( 2 - √3 )

= ( 2 - √3 ) / { ( 2 )² - ( √3 )² }           [ ( a + b ) ( a - b ) = a² - b² ]

= ( 2 - √3 ) / ( 4 - 3 )

= ( 2 - √3 ).

For , 2 / ( √5 - √3 ) ,

To rationalize the denominator we shall multiply the numerator and denominator both by ( √5 + √3 ),

= 2 ( √5 + √3 ) / ( √5 - √3 ) ( √5 + √3 )

= 2 ( √5 + √3 ) / { ( √5 )² - ( √3 )² }          [ ( a + b ) ( a - b ) = a² - b² ]

= 2 ( √5 + √3 ) / ( 5 - 3 )

= 2 ( √5 + √3 ) / 2 = ( √5 + √3 )

For, 1 / ( 2 - √5 ) ,

We shall multiply the numerator and denominator by ( 2 + √5 ) to rationalize its denominator.

= 1 ( 2 + √5 ) / ( 2 - √5 ) ( 2 + √5 )

= ( 2 + √5 ) / { ( 2 )² - ( √5 )² }     [ ( a + b ) ( a - b ) = a² - b² ]

= ( 2 + √5 ) / ( 4 - 5 )

= ( 2 + √5 ) / ( - 1 )

= - ( - 2 - √5 ) / ( - 1 )

= ( - 2 - √5 )

Now, by adding all these ,

 = 2 - √3 + √5 + √3 - 2 - √5

= 0 . Proved.




                                                       ☺☺☺