Expand `(a-4)^(3)`
A. `a^(3)-12a^(2) +48a +64`
B. `a^(3) -48a^(2) +12 a -64`
C. `a^(3) +12 a^(2) -48a -64`
D. `a^(3) -12 a^(2) + 48a -64`

Options

A. C
B. A
C. D
D. B

Answer

To expand \((a - 4)^3\), we can use the binomial expansion formula or the identity for the cube of a binomial. The identity we will use is:

\[ (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \]

In this case, \(a\) is \(a\) and \(b\) is \(4\).

### Step-by-Step Solution:

1. **Identify \(a\) and \(b\)**:
\[ a = a, \quad b = 4 \]

2. **Apply the identity**:
\[ (a - 4)^3 = a^3 - 3a^2(4) + 3a(4^2) - 4^3 \]

3. **Calculate each term**:
- First term: \(a^3\)
- Second term: \(-3a^2(4) = -12a^2\)
- Third term: \(3a(4^2) = 3a(16) = 48a\)
- Fourth term: \(-4^3 = -64\)

4. **Combine all the terms**:
\[ (a - 4)^3 = a^3 - 12a^2 + 48a - 64 \]

Thus, the expanded form of \((a - 4)^3\) is:
\[ a^3 - 12a^2 + 48a - 64 \]

### Final Answer:
The correct option is **D**: \(a^3 - 12a^2 + 48a - 64\).