To find the new volume of the cube after its side lengths are reduced, we can start with the original side length of the cube, which is 2 units.
The original volume of the cube is given by:
\[ V = s^3 \]
where \( s \) is the side length of the cube. For \( s = 2 \),
\[ V_{\text{original}} = 2^3 = 8. \]
Now, we want to reduce the side length by 3 units. The new side length \( s' \) will be:
\[ s' = 2 - 3 = -1. \]
Since the side length cannot be negative, this initially seems problematic. Instead, we should reinterpret the meaning of the problem. The reduction of 3 units cannot be applied directly here since the side length cannot be less than zero. We can infer that this might imply a change in the perspective of cube dimensions for any valid size.
Instead, let’s consider the term "reduce by 3 units" more generally in relation to calculating the volume using the binomial theorem. For any side length expressed as \( s = 2 \), and \( x = -3 \):
We can express the side length after applying a reduction of 3 units generally as:
\[ s' = 2 +(-3) = 2 + x. \]
In this case \( x = -3 \). We can now use the binomial theorem to express \( (s + x)^3 \).
Using the binomial theorem: \[ (s + x)^3 = s^3 + 3s^2x + 3sx^2 + x^3. \] Substituting \( s = 2 \) and \( x = -3 \):
\[ (2 - 3)^3 = 2^3 + 3(2^2)(-3) + 3(2)(-3)^2 + (-3)^3. \]
Now calculating each term:
- The first term: \( 2^3 = 8 \)
- The second term: \( 3(2^2)(-3) = 3(4)(-3) = -36 \)
- The third term: \( 3(2)(-3)^2 = 3(2)(9) = 54 \)
- The fourth term: \( (-3)^3 = -27 \)
Now, summing these contributes to the new volume:
\[ V' = 8 - 36 + 54 - 27. \]
Calculating this step-by-step: \[ V' = 8 - 36 = -28, \] \[ -28 + 54 = 26, \] \[ 26 - 27 = -1. \]
Thus, the new volume of the cube, albeit mathematically computed, leads us to conclude a non-physical negative volume. However, that considered mathematically you will see:
\[ \text{The correct expression for the new volume using the binomial theorem is} \ (2 - 3)^3 = -1 \text{ (implying the cube is physically invalid with negative side)}. \]
This illustrates \( (s - 3 )^3 \) evidently leading to reinterpretation discussing dimensions and requirement in scenarios beyond zero. Hence if required rethink practical dimensional setups or side limits.