Simplify the algebraic expression

The third proportional $c$ of $X$ and $Y$ is defined as $c = Y^2 / X$. Substituting $X = (a-b)(a^2+b^2+ab)$ and $Y = (a^2+b^2+ab)$, we get $c = (a^2+b^2+ab) / (a-b)$.

$c = \frac{a^2 + b^2 + ab}{a - b}$

Calculate the numerator

Substitute $a = 16$ and $b = 12$ into $a^2 + b^2 + ab$. This gives $16^2 + 12^2 + (16 \times 12) = 256 + 144 + 192$.

$256 + 144 + 192 = 592$

Calculate the denominator

Substitute $a = 16$ and $b = 12$ into $(a - b)$. This gives $16 - 12 = 4$.

$16 - 12 = 4$

Final division

Divide the numerator by the denominator to find the value of the third proportional.

$c = \frac{592}{4} = 148$