To determine the volume of a cube with a side length of $4 \, \text{cm}$, we can apply the formula:

$V = a^3$

where $V$ represents the volume and $a$ is the length of one side of the cube. The volume of a cube is calculated by raising the length of one of its sides to the power of three. This is due to the nature of a cube, which has equal lengths for all its sides. Therefore, the volume is the product of its length, width, and height, all of which are identical.

In this case, the side length $a$ is $4 \, \text{cm}$.

To find the volume, we multiply $4 \, \text{cm}$ by itself three times:

$V = 4 \, \text{cm} \times 4 \, \text{cm} \times 4 \, \text{cm}$

This expression can also be simplified to:

$V = 4^3 \, \text{cm}^3$

Next, we calculate $4^3$ as follows:

$4 \times 4 = 16$

Then we multiply this result by $4$ again:

$16 \times 4 = 64$

Thus, we find that the volume of the cube is:

$V = 64 \, \text{cm}^3$

This indicates that the cube has a capacity of $64$ cubic centimeters. It is important to remember that the unit for volume is always expressed in cubic units. In this case, we use cubic centimeters ($\text{cm}^3$), as volume measures three-dimensional space.