What is the minimum value of $({ a }^{ 3 }{ c }^{ 2 }+{ a }^{ 2 }{ b }^{ 3 }+{ b }^{ 2 }{ c }^{ 3 })({ b }^{ 3 }c+a{ c }^{ 3 }+{ a }^{ 3 }b)$ if the value of ${ a }^{ 3 }{ b }^{ 3 }{ c }^{ 3 }=27$ and given that they all are positive real numbers.

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The answer is 243.

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since $a^3b^3c^3=27$ , $b^3c+ac^3+a^3b=\dfrac{b^3a^3c^3}{a^3c^2}+\dfrac{b^3a^3c^3}{a^2b^3}+\dfrac{b^3a^3c^3}{b^2c^3}=\dfrac{27}{a^3c^2}+\dfrac{27}{a^2b^3}+\dfrac{27}{b^2c^3}$ by cauchy sarwaz: $({ a }^{ 3 }{ c }^{ 2 }+{ a }^{ 2 }{ b }^{ 3 }+{ b }^{ 2 }{ c }^{ 3 })({ b }^{ 3 }c+a{ c }^{ 3 }+{ a }^{ 3 }b)=({ a }^{ 3 }{ c }^{ 2 }+{ a }^{ 2 }{ b }^{ 3 }+{ b }^{ 2 }{ c }^{ 3 })(\dfrac{27}{a^3c^2}+\dfrac{27}{a^2b^3}+\dfrac{27}{b^2c^3})\geq (\sqrt{27}+\sqrt{27}+\sqrt{27})^2=\boxed{243}$