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Rationalize the denominator of . With your answer in the form , and the fraction in lowest terms, what is A+B+C+D?

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Rationalize the denominator of $$\displaystyle \frac{1}{\sqrt[3]{3} - \sqrt[3]{2}}$$  . 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?

tertre  May 29, 2017
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#1
+75356
+2

We can write

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

Multiply top/bottom  by   [ 3^(2/3) +  (3*2)^(1/3) + 2^(2/3) ]

And we have

[ 3^(2/3) + (3*2)^(1/3) + 2^(2/3) ]  /  [ 3 + (18)^(1/3) + (12)^(1/3) - (18)^(1/3) - 12^(1/3) - 2 ]

[ (9)^(1/3)  + (6)^(1/3) + (4)^(1/3) ]  /  1

So   A + B + C + D    =  9 + 6 + 4 + 1  =   20

CPhill  May 29, 2017

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