To find the value of the expression \(a^3 - 3a^2b + 3ab^2 - b^3\) for \(a = 2\) and \(b = 4\), follow these steps:
1. Substitute the values of \(a\) and \(b\) into the expression:
[tex]\[
2^3 - 3 \cdot 2^2 \cdot 4 + 3 \cdot 2 \cdot 4^2 - 4^3
\][/tex]
2. Calculate each term separately:
- Compute \(2^3\):
[tex]\[
2^3 = 8
\][/tex]
- Compute \(3 \cdot 2^2 \cdot 4\):
[tex]\[
2^2 = 4 \quad \text{so} \quad 3 \cdot 4 \cdot 4 = 3 \cdot 16 = 48
\][/tex]
- Compute \(3 \cdot 2 \cdot 4^2\):
[tex]\[
4^2 = 16 \quad \text{so} \quad 3 \cdot 2 \cdot 16 = 6 \cdot 16 = 96
\][/tex]
- Compute \(4^3\):
[tex]\[
4^3 = 64
\][/tex]
3. Substitute these results back into the expression:
[tex]\[
8 - 48 + 96 - 64
\][/tex]
4. Simplify the expression step-by-step:
- First add and subtract from left to right:
[tex]\[
8 - 48 = -40
\][/tex]
Then add the next term:
[tex]\[
-40 + 96 = 56
\][/tex]
Finally, subtract the last term:
[tex]\[
56 - 64 = -8
\][/tex]
So, the value of the expression \(a^3 - 3a^2b + 3ab^2 - b^3\) when \(a = 2\) and \(b = 4\) is:
[tex]\(\boxed{-8}\)[/tex]
