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Simplify \frac{4}{1 - \sqrt[3]{3}} + \sqrt[3]{9}.

 Jun 19, 2024
 #1
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\(\frac{4}{1 - \sqrt[3]{3}} + \sqrt[3]{9}. \)

 

We can write  (1 - 3)  as  ( 1 - 3^(1/3))  ( 1 + 3^(1/3) + 3^(2/3) )

 

Mulitiplying the numerator and  denominator of the first fraction by the last factor in red we get

 

4 [ 1 + 3^(1/3) + 3^(2/3) ]          -2 [ 1 + 3^(1/3) + 3^(2/3) ]

____________________  =  

               1 - 3

 

And  9^(1/3)  = ( 3^2)^(1/3)  =  3^(2/3)

 

So we have

 

 -2 [ 1 + 3^(1/3) + 3^(2/3) ] + 3^(2/3)   =

 

-2  - 2*3^(1/3) - 3^(2/3)

 

cool cool cool

 Jun 20, 2024

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