Find the value of $(52+6\sqrt{43})^{3/2}-(52-6\sqrt{43})^{3/2}$.
Suppose that $52+6\sqrt{43}$ is in the form of $(a + b\sqrt{43})^2$. [FOILing](https://artofproblemsolving.com/wiki/index.php/FOIL) yields that $52 + 6\sqrt{43} = a^2 + 43b^2 + 2ab\sqrt{43}$. This implies that $a$ and $b$ equal one of $\pm3, \pm1$. The possible [sets](https://artofproblemsolving.com/wiki/index.php/Set) are $(3,1)$ and $(-3,-1)$; the latter can be discarded since the [square root](https://artofproblemsolving.com/wiki/index.php/Square_root) must be positive. This means that $52 + 6\sqrt{43} = (\sqrt{43} + 3)^2$. Repeating this for $52-6\sqrt{43}$, the only feasible possibility is $(\sqrt{43} - 3)^2$.
Rewriting, we get $(\sqrt{43} + 3)^3 - (\sqrt{43} - 3)^3$. Using the difference of [cubes](https://artofproblemsolving.com/wiki/index.php/Cube), we get that $[\sqrt{43} + 3\ - \sqrt{43} + 3]\ [(43 + 6\sqrt{43} + 9) + (43 - 9) + (43 - 6\sqrt{43} + 9)]$ $= (6)(3 \cdot 43 + 9) = \boxed{828}$.
Note: You can also just use the formula $(a + b)^2 = a^2 + 2ab + b^2$ instead of [ foiling](https://artofproblemsolving.com/wiki/index.php/FOIL).