Bráulia cut a cube into many smaller cubes with an edge of $1 \mathrm{~cm}$, through cuts parallel to its faces. For example, if this cube had a side of $4 \mathrm{~cm}$, the cuts would produce:

![](https://cdn.mathpix.com/cropped/2024_05_01_d18780f0fd52e0502cf8g-02.jpg?height=360&width=374&top_left_y=551&top_left_x=744)

However, the length of the edge of this cube is unknown.

a) After cutting the cube, Bráulia counted the $1 \mathrm{~cm}$ side smaller cubes, which were 512. What was the length of the edge of the cube?

b) Laura does the same with another cube, for which the length of the edge is also unknown. After the cut, Laura counts 512 smaller cubes that previously had no face in contact with the exterior of the cube. What was the length of the edge of the cube?
Solution

a) Since $512=8^{3}$, the length of the edge of the cube before it was cut was $8 \mathrm{~cm}$.

b) The small cubes that had no face in contact with the exterior of the original cube also formed a cube. Since 512 such small cubes were counted, this means that the inner cube had an edge of $8 \mathrm{~cm}$ (using the answer from the previous item). The edge of the inner cube is two units smaller than that of the original cube. Therefore, the length of the edge of the original cube was $10 \mathrm{~cm}$.