# How do I rationalize the following fraction: 1/(9root[3](9)-3root[3](3)-27)?

solvarmedw 2022-10-02 Answered
How do I rationalize the following fraction: $\frac{1}{9\sqrt[3]{9}-3\sqrt[3]{3}-27}$?
As the title says I need to rationalize the fraction: $\frac{1}{9\sqrt[3]{9}-3\sqrt[3]{3}-27}$. I wrote the denominator as: $\sqrt[3]{{9}^{4}}-\sqrt[3]{{9}^{2}}-{3}^{3}$ but I do not know what to do after. Can you help me?
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## Answers (1)

Marcel Mccullough
Answered 2022-10-03 Author has 11 answers
We have
$\left(x+y+z\right)\left({x}^{2}+{y}^{2}+{z}^{2}-xy-xz-yz\right)={x}^{3}+{y}^{3}+{z}^{3}-3xyz$
With $x=9\sqrt[3]{9},y=-3\sqrt[3]{3},z=-27$ all terms on the right side are rational, try it. So multiply the given numerator and denominator by ${x}^{2}+{y}^{2}+{z}^{2}-xy-xz-yz$ with $x,y,z$ as rendered above.
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