3. Place a regular tetrahedron with volume 1 inside a cube, then the minimum volume of this cube is $\qquad$
3. 3 .

Considering in reverse, for a cube with edge length $a$ (volume $a^{3}$), the largest inscribed regular tetrahedron has vertices formed by the cube's vertices that do not share an edge, and its volume is $\frac{a^{3}}{3}$.
$$
\text { Let } \frac{a^{3}}{3}=1 \text {, then } a^{3}=3 \text {. }
$$