If `( a + 4) ^(3) = a^(3) + 12 a^(2) + ka + 64` then what is the value of k ?

Options

A. 12
B. 24
C. 36
D. 48

Answer

To find the value of \( k \) in the equation

\[ (a + 4)^3 = a^3 + 12a^2 + ka + 64, \]

we can start by expanding the left-hand side using the binomial theorem or the formula for the cube of a binomial.

### Step 1: Expand \( (a + 4)^3 \)

Using the formula for the cube of a binomial, which states that

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

we can set \( a = a \) and \( b = 4 \):

\[ (a + 4)^3 = a^3 + 3a^2(4) + 3a(4^2) + 4^3. \]

Calculating each term:

- \( 3a^2(4) = 12a^2 \)
- \( 3a(4^2) = 3a(16) = 48a \)
- \( 4^3 = 64 \)

Putting it all together, we get:

\[ (a + 4)^3 = a^3 + 12a^2 + 48a + 64. \]

### Step 2: Set the expanded form equal to the right-hand side

Now we can equate the expanded form to the right-hand side of the original equation:

\[ a^3 + 12a^2 + 48a + 64 = a^3 + 12a^2 + ka + 64. \]

### Step 3: Cancel out the common terms

We can cancel \( a^3 \), \( 12a^2 \), and \( 64 \) from both sides:

\[ 48a = ka. \]

### Step 4: Solve for \( k \)

Since \( a \) is not equal to zero (assuming \( a \neq 0 \)), we can divide both sides by \( a \):

\[ 48 = k. \]

Thus, the value of \( k \) is

\[ \boxed{48}. \]

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