Question 1179366
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See this:
https://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means

and

https://math.stackexchange.com/questions/1298115/prove-inequality-abbcca-ge-3-abc-1

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If you expand the left hand side you will get:

    {{{ matrix(3,3, "", "" , "", "", (x + (abc)^(1/3))^3, "","","","")}}} = {{{ matrix(3,3,"","","","", x^3 + 3(abc)^(1/3)x^2 + 3(abc)^(2/3)x + abc,"","","","") }}}

Expanding the middle gives:
    {{{ (x+a)(x+b)(x+c)}}} = {{{x^3 + (a+b+c)x^2 + (ab+ac+bc)x + abc }}}

Comparing term-by term, you get two inequalities to prove:
     {{{ matrix(3,3, "","","","", 3(abc)^(1/3) <= a+b+c,"","","","") }}}
     {{{ matrix(3,3,"","","","", 3(abc)^(2/3) <= ab+bc+ac,"","","","") }}}

The LHS is a GM, the RHS is an AM,  so here the AM-GM inequality applies.

The middle term compared to the RHS (in original problem statement) is then proven in a similar fashion.