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Simplify and rationalize the denominator:\(\frac{1}{1+ \frac{1}{\sqrt{3}+1}}.\)

 Dec 4, 2019
 #1
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It's easiest to work just with the fraction in the denominator, first....so we have

 

  1  / [  √3 + 1]     [ multiply top/bottom by  √3 - 1 ]  =

 

[  √3 - 1]  /  [3 - 1]  =

 

[  √3 - 1 ] /  2

 

So....on the bottom we have

 

1  +  [  √3 - 1 ] / 2    =

 

[ 2 +  √3 - 1 ]  / 2  =

 

[  √3 + 1 ]  / 2

 

So we have

 

   1

___________  =

[ √3 + 1 ] / 2

 

    2

_________         again, multiply top/bottom by  √3 - 1   

[  √3 + 1 ]

 

 

2 [  √3 - 1 ]

__________  =

   3   -  1

 

 

2 [  √3 -1]

__________  =

       2 

 

 √3 - 1

 

cool cool cool

 Dec 4, 2019

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