Irrationality In Big Sums

The denominator looks like the second factor of the difference of two cubes identity $x^3 - y^3 = (x-y)(x^2+xy+y^2)$ with $x= \sqrt [ 3 ]{ k+1 }$ and $y = \sqrt [ 3 ]{ k }$ . Indeed, from $\sqrt [ 3 ]{ k+1 }^{ 3 } - \sqrt [ 3 ]{ k }^{ 3 } = ( \sqrt [ 3 ] { k+1 } - \sqrt [ 3 ]{ k}) ( \sqrt [ 3 ]{ { k }^{ 2 } } + \sqrt [ 3 ]{ k(k + 1) } + \sqrt [ 3 ]{ { ( k + 1 ) }^{ 2 } } )$

it directly follows that

$\frac { 1 }{ \sqrt [ 3 ]{ { k }^{ 2 } } + \sqrt [ 3 ]{ k(k + 1) } + \sqrt [ 3 ]{ { \left( k + 1 \right) }^{ 2 } } }= \frac{\sqrt[3]{k+1}-\sqrt[3]{k}}{\sqrt[3]{(k+1)^{3}}-\sqrt[3]{k^{3}}} = \frac{\sqrt[3]{k+1}-\sqrt[3]{k}}{k+1-k}=\sqrt[3]{k+1}-\sqrt[3]{k}$ .

Thus, this sum telescopes to $\sqrt [ 3 ]{ 2197 } - \sqrt [3]{ 1 } = 13-1 = 12.$