That's too long!

Let $x = \left(27+\sqrt{756}\right)^\frac 13 + \left(27-\sqrt{756}\right)^\frac 13$ . Then we have:

$\begin{aligned} x^3 & = \left( \left(27+\sqrt{756}\right)^\frac 13 + \left(27-\sqrt{756}\right)^\frac 13 \right)^3 \\ & = \color{#3D99F6}{27+\sqrt{756}} + 3 \left(27+\sqrt{756}\right)^\frac 23 \left(27-\sqrt{756}\right)^\frac 13 + 3 \left(27+\sqrt{756}\right)^\frac 13 \left(27-\sqrt{756}\right)^\frac 23 + \color{#3D99F6}{27-\sqrt{756}} \\ & = \color{#3D99F6}{54} + 3 \left(27^2 - 756 \right)^\frac 13 \left( \left(27+\sqrt{756} \right)^\frac 13 + \left(27-\sqrt{756}\right)^\frac 13 \right) \\ & = 54 + 3 \left(- 27 \right)^\frac 13 x \\ & = 54 - 9x \end{aligned}$

$\begin{aligned} \implies x^3 + 9x - 54 & = 0 \\ (x-3)(x^2+3x+18) & = 0 \\ \implies x & = \boxed{3} \end{aligned}$

Note that $x^2 + 3x + 18 = 0$ has no real root.