Algebra

First, let's zoom in and focus on the equation we are given. Let's square it. We get

$(a^2 - 2 + 1/a^2) = 0$

This will achieve the equation

$a^2 - 1/a^2 = -2$

Next, let's focus on what we are trying to find. We know that

$a^3 - 1/a^3 = ( a - 1/a) ( a^2 + 1 + 1/a^2)$

WAIT! The first term in what we must find is 0!

That means our answer is just 0. 

So 0 is our answer!

Thanks! :)