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Find $\displaystyle{ \frac{2}{1 + 2\sqrt{3}} + \frac{3}{2 - \sqrt{3}}}$, 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 $A+B+C$?

 Jan 13, 2018
 #1
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Convert   :\( $\displaystyle{ \frac{2}{1 + 2\sqrt{3}} + \frac{3}{2 - \sqrt{3}}}$\)

 

To the form  : \( $\displaystyle \frac{A + B\sqrt{3}}{C}$\)

 

Where  \($A > 0$\)

 

And find  \($A+B+C$?\)

 

 

[  2 * (2 - √3)  +  3 ( 1 + 2√3)  ]  /  [  ( 1 + 2 √3)  ( 2 - √3) ]  =

 

[ 4 - 2√3+ 3 + 6√3 ]  /  [ 2 + 4√3 - √3 - 6 ]   =

 

[   7 + 4√3 ]  /  [ 3√3 - 4]

 

[   7 + 4√3 ]  *  [ 3√3 + 4]   /   [  ( 3√3 - 4) (3√3 + 4 ) ]

 

[ 21√3 + 36 + 28 + 16√3]  / [ 27 - 16 ]  

 

[ 64 + 37√3]  /  11

 

So.....  A  = 64, B = 37  , C  =  11

 

So

 

A + B + C  =     112

 

 

cool cool cool

 Jan 13, 2018
edited by CPhill  Jan 13, 2018

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