Simple surds?

In general we have that $(a - b)^{3} = a^{3} - 3a^{2}b + 3ab^{2} - b^{3} = a^{3} - 3ab(a - b) - b^{3}.$

Now let $\large a = \sqrt[3]{9 + 4\sqrt{5}}$ , $\large b = \sqrt[3]{9 - 4\sqrt{5}}$ and $x = a - b.$

Then $a^{3} - b^{3} = (9 + 4\sqrt{5}) - (9 - 4\sqrt{5}) = 8\sqrt{5}$ and $ab = \sqrt[3]{(9 + 4\sqrt{5})(9 - 4\sqrt{5})} = \sqrt[3]{81 - 16*5} = 1.$

We thus have that $x^{3} = 8\sqrt{5} - 3x \Longrightarrow x^{3} + 3x = 8\sqrt{5}.$

Squaring both sides of this equation yields $x^{6} + 6x^{4} + 9x^{2} = 64*5 = 320,$ which after letting $S = x^{2}$ becomes

$S^{3} + 6S^{2} + 9S - 320 = 0.$

Checking the divisors of $320$ we see that $S = 5$ is a solution, and so the equation can be factored as

$(S - 5)(S^{2} + 11S + 64) = 0.$

Now as the discriminant of $S^{2} + 11S + 64$ is $11^{2} - 4*64 = -135 \lt 0$ we can conclude that

$x^{2} = S = \boxed{5}$ is the unique solution.