Find the volume of the cuboid in the form $a\sqrt{b} \text{ cm}^3$, where $a, b \in \mathbb{N}$. - Leaving Cert Mathematics - Question 3(a) - 2021

From the equation $xz = 2\sqrt{2}$, we can express $z$ in terms of $x$ as follows:

$z = \frac{2\sqrt{2}}{x}$

Substituting into the second equation, we can express $y$ in terms of $x$:

$y \left(\frac{2\sqrt{2}}{x}\right) = 8\sqrt{6} \Rightarrow y = \frac{8\sqrt{6}x}{2\sqrt{2}} = 4\sqrt{3}x$

Now, substituting $y$ and $z$ into the formula for volume $V = xyz$:

$V = x(4\sqrt{3}x)\left(\frac{2\sqrt{2}}{x}\right) = 8\sqrt{6}x$

Now substituting $x = 1$ (since we have used $x^2 = 1$ from other calculations):

$V = 8\sqrt{6} \text{ cm}^3$