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Find $\displaystyle{ \frac{2}{1 + 2\sqrt{3}} + \frac{5}{2 - \sqrt{3}}}$, and write your answer in the form $\displaystyle \frac{A + B\sqrt{3}}{C}$, with the fraction in lowest terms and $A > 0$. What is $A+B+C$?

 Apr 29, 2022
 #1
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\( $\displaystyle{ \frac{2}{1 + 2\sqrt{3}} + \frac{5}{2 - \sqrt{3}}}$\)

 

 

We have

 

2 [ 2 - sqrt 3 ] + 5 [ 1 + 2sqrt 3 ]

_________________________   =

  (1 + 2sqrt 3)  ( 2 - sqrt 3)

 

 

4 - 2sqrt 3  + 5 + 10sqrt 3

______________________  =

  2  + 4sqrt 3 - sqrt 3  - 2*3

 

9 + 8sqrt 3

_________   =     

-4 + 3sqrt 3

 

(9  + 8sqrt 3) ( -4 -3sqrt 3)

______________________  =

(-4 + 3sqrt 3) ( -4 - 3sqrt 3)

 

-36 - 32sqrt 3 - 27sqrt 3 - 24*3

__________________________  =

16 - 9*3

 

-108 - 59sqrt 3

___________  =

       -11

 

108 + 59sqrt 3

___________

         11

 

A  +  B  +  C  = 

 

108  + 59  + 11   = 

 

178

 

cool cool cool

 Apr 30, 2022

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