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Rationalize the denominator of \(\displaystyle \frac{1}{\sqrt[3]{3} - 1}\). With your answer in the form \(\displaystyle \frac{\sqrt[3]{A} + \sqrt[3]{B} + \sqrt[3]{C}}{D}\) , and the fraction in lowest terms, what is A + B + C +D ?

 Jul 13, 2022
 #1
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Multiply top / bottom  by     1 + 3^(1/3)  + 3^(2/3)   

 

Note that the bottom becomes

 

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

 

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

 

So  we have

 

 

1 + 3^(1/3)  + 3^(2/3)             ∛1  + ∛3 + ∛9

__________________  =     _____________

                 2                                    2

 

A + B + C + D   =   15

 

 

cool cool cool

 Jul 13, 2022

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