Express `(4-3i)^(3)` in the form `(a+ib)`.

Answer

To express \((4 - 3i)^3\) in the form \(a + ib\), we will use the formula for the cube of a binomial, which is given by:

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

### Step 1: Identify \(a\) and \(b\)
In our case, we have:
- \(a = 4\)
- \(b = 3i\)

### Step 2: Apply the formula
Using the formula, we can expand \((4 - 3i)^3\):

\[ (4 - 3i)^3 = 4^3 - (3i)^3 - 3(4)(3i)(4 - 3i) \]

### Step 3: Calculate \(4^3\) and \((3i)^3\)
Calculating \(4^3\):
\[ 4^3 = 64 \]

Calculating \((3i)^3\):
\[ (3i)^3 = 27i^3 = 27(-i) = -27i \]

### Step 4: Calculate \(3(4)(3i)\)
Calculating \(3(4)(3i)\):
\[ 3(4)(3i) = 36i \]

### Step 5: Substitute back into the expression
Now substituting these values back into the equation:

\[ (4 - 3i)^3 = 64 - (-27i) - 36i(4 - 3i) \]

### Step 6: Simplify the expression
Now we simplify the expression:

\[ = 64 + 27i - 36i(4 - 3i) \]

Calculating \(36i(4 - 3i)\):
\[ 36i \cdot 4 = 144i \]
\[ 36i \cdot (-3i) = -108i^2 = 108 \quad (\text{since } i^2 = -1) \]

So,
\[ 36i(4 - 3i) = 144i + 108 \]

### Step 7: Substitute and combine like terms
Now substituting this back into our expression:

\[ (4 - 3i)^3 = 64 + 27i - (144i + 108) \]

Combining like terms:
\[ = 64 - 108 + (27i - 144i) \]
\[ = -44 - 117i \]

### Final Result
Thus, we can express \((4 - 3i)^3\) in the form \(a + ib\) as:

\[ (4 - 3i)^3 = -44 - 117i \]