Doubling the cube
= 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.