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Let a,b and c be nonzero real numbers such that a/b + b/c + c/a =7 and b/a + c/b + a/c = 9. 

 

Find a^3/b^3 + b^3/c^3 + c^3/a^3. 
 

 Aug 29, 2019
 #1
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Let a,b and c be nonzero real numbers such that a/b + b/c + c/a =7 and b/a + c/b + a/c = 9. 

Find a^3/b^3 + b^3/c^3 + c^3/a^3.

 

\(\begin{array}{|rcll|} \hline && \mathbf{\left( \dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{a} \right)^3 -3 \left(\dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{a} \right) \left(\dfrac{b}{a} + \dfrac{c}{b} + \dfrac{a}{c}\right )} \\\\ &=& \dfrac{a^3}{b^3} + \dfrac{c^3}{a^3} + \dfrac{3a^2}{bc} + \dfrac{3bc}{a^2} + \dfrac{3ac}{b^2} + \dfrac{3b^2}{ac} + \dfrac{3ab}{c^2} + \dfrac{3c^2}{ab} + \dfrac{b^3}{c^3} + 6 \\ && - \left(\dfrac{3a^2}{b c} + \dfrac{3bc}{a^2} + \dfrac{3ac}{b^2} + \dfrac{3b^2}{ac} + \dfrac{3ab}{c^2} + \dfrac{3c^2}{a b} + 9\right) \\\\ &=& \mathbf{\dfrac{a^3}{b^3} + \dfrac{b^3}{c^3} + \dfrac{c^3}{a^3} - 3} \\ \hline \end{array}\)

 

 

\(\begin{array}{|rcll|} \hline \mathbf{\dfrac{a^3}{b^3} + \dfrac{b^3}{c^3} + \dfrac{c^3}{a^3} - 3 } &=& \mathbf{\left( \dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{a} \right)^3 -3 \left(\dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{a} \right) \left(\dfrac{b}{a} + \dfrac{c}{b} + \dfrac{a}{c}\right )} \\\\ \dfrac{a^3}{b^3} + \dfrac{b^3}{c^3} + \dfrac{c^3}{a^3} &=& 3+ \left( \dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{a} \right)^3 -3 \left(\dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{a} \right) \left(\dfrac{b}{a} + \dfrac{c}{b} + \dfrac{a}{c}\right ) \\\\ \dfrac{a^3}{b^3} + \dfrac{b^3}{c^3} + \dfrac{c^3}{a^3} &=& 3+ \left( 7 \right)^3 -3 \left(7 \right) \left(9\right ) \\\\ &=& 3+ 343 - 189 \\\\ \mathbf{\dfrac{a^3}{b^3} + \dfrac{b^3}{c^3} + \dfrac{c^3}{a^3} } &=& \mathbf{157} \\ \hline \end{array}\)

 

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 Aug 30, 2019

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