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Simplify \(\frac{2\sqrt[3]9}{1 + \sqrt[3]3 + \sqrt[3]9}. \)

 Jul 10, 2018
 #1
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Simplify 

\(\displaystyle \frac{2\sqrt[3]9}{1 + \sqrt[3]3 + \sqrt[3]9}.\)

\frac{2\sqrt[3]9}{1 + \sqrt[3]3 + \sqrt[3]9}.

 

Geometric progression: \(1 + \sqrt[3]3 + \sqrt[3]9 \qquad a = 1 \text{ and } r = \sqrt[3]3 \)

Formula: \(1+r+r^2 = \dfrac{r^3-1}{r-1}\)

\(\begin{array}{|rcll|} \hline 1 + \sqrt[3]3 + \sqrt[3]9 &=& \dfrac{(\sqrt[3]3)^3-1}{\sqrt[3]3-1} \\\\ &=& \dfrac{3-1}{\sqrt[3]3-1} \\\\ &=& \dfrac{2}{\sqrt[3]3-1} \\ \hline \end{array}\)

 

\(\begin{array}{|rcll|} \hline && \mathbf{ \dfrac{2\sqrt[3]9}{1 + \sqrt[3]3 + \sqrt[3]9} } \\\\ &=& \dfrac{2\sqrt[3]9}{\dfrac{2}{\sqrt[3]3-1}} \\\\ &=& \sqrt[3]9(\sqrt[3]3-1) \\\\ &=& \sqrt[3]{3^3}-\sqrt[3]9 \\\\ &\mathbf{=}& \mathbf{3-\sqrt[3]9} \\ \hline \end{array}\)

 

 

laugh

 Jul 10, 2018

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