Solution:
Assuming the square has side length $x$, it has area $x^{2}$, so the equilateral triangle has side length $x^{2} / 3$. The area of the equilateral triangle is then given in two ways by
$$4x = \frac{\sqrt{3}}{4} \left(\frac{x^{2}}{3}\right)^{2}.$$
Solving gives $x^{3} = \frac{144}{\sqrt{3}} = 48 \sqrt{3}$. Then $x^{6} = 48^{2} \cdot 3$ and $x^{2} = \sqrt[3]{48^{2} \cdot 3} = \sqrt[3]{2^{8} 3^{3}} = 12 \sqrt[3]{4}$