Doubling the cube

Doubling the cube is the classical construction task of producing a cube with twice the volume of a given cube. If the original edge has length $a>0$, the new edge must have length $a\sqrt[3]{2}$, since volume scales as the cube of edge length.

Starting from a unit segment, exact straightedge and compass construction cannot produce this length. The polynomial $x^3-2$ is irreducible over $\mathbb{Q}$ by the rational root theorem, so $[\mathbb{Q}(\sqrt[3]{2}):\mathbb{Q}]=3$. A real constructible number lies in a tower of quadratic extensions, and its degree over $\mathbb{Q}$ must therefore be a power of two. Three fails that necessary condition.

This is an impossibility for the stated construction tools, not for approximation or every enlarged tool set. A decimal approximation can be made as accurate as desired; allowing additional construction operations changes which lengths are exactly obtainable.