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The number (\sqrt{2}+\sqrt{3})^3$ can be written in the form $a\sqrt{2} + b\sqrt{3} + c\sqrt{6}$, where $a$, $b$, and $c$ are integers. What is $a+b+c$?

 May 22, 2019
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\( (\sqrt{2}+\sqrt{3})^3\)

 

Using the binomial expansion we have

 

( √2)^3  + 3 ( √ 2)^2 (√3)  + 3( √2)( √3)^2  + ( √3)^3  =

 

2 √2  + 6 √3  +  9 √2  +  3 √3  =

 

11 √2  + 9 √3  + 0 √6

 

So  a + b + c   =   11 + 9 + 0    =  20

 

 

cool cool cool

 May 22, 2019

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