$\large 4(a^6+b^6+c^6)+5abc(a^3+b^3+c^3)$

Given that $a$ , $b$ and $c$ are reals satisfying $ab+bc+ca=\sqrt[3]{100}$ .

Find the minimum value of the expression above.

The answer is 100.

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Symmetric expressions always attain minimum values when its variables are all equal.

Thus for the minimum value we can take $a=b=c$ giving us $3a^2=100^⅓$ and the minimum value of the expression as

100.