For all real number x and the imaginary number i, which of the following expressions is equivalent to `(x - 3i)^(3)`?

Options

A. `x^(3) - 9x^(2)i -27x + 27i`
B. `x^(3) + 9x^(2)i -27x - 27i`
C. `x^(3) + 3x^(2)i -9x - 27i`
D. `x^(3) - 3x^(2)i -9x + 27i`

Answer

To solve the expression \((x - 3i)^3\), we can use the identity for the cube of a binomial, which states:

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

### Step 1: Identify \(a\) and \(b\)
Here, we identify:
- \(a = x\)
- \(b = 3i\)

### Step 2: Apply the identity
Using the identity, we can substitute \(a\) and \(b\):

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

### Step 3: Calculate \((3i)^3\)
Now, we calculate \((3i)^3\):

\[ (3i)^3 = 27i^3 \]

Since \(i^2 = -1\), we know that \(i^3 = i \cdot i^2 = i \cdot (-1) = -i\). Therefore:

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

### Step 4: Substitute back into the expression
Now we substitute this back into our equation:

\[ (x - 3i)^3 = x^3 - (-27i) - 3(x)(3i)(x - 3i) \]

This simplifies to:

\[ (x - 3i)^3 = x^3 + 27i - 9xi(x - 3i) \]

### Step 5: Expand \(-9xi(x - 3i)\)
Now we expand \(-9xi(x - 3i)\):

\[ -9xi(x - 3i) = -9xi^2 + 27x \]

Since \(i^2 = -1\), we have:

\[ -9xi^2 = 9x \]

Thus, the expression becomes:

\[ (x - 3i)^3 = x^3 + 27i - 9x + 27x \]

### Step 6: Combine like terms
Combining the terms, we get:

\[ (x - 3i)^3 = x^3 - 9x + 27i \]

### Final Expression
Thus, the final expression for \((x - 3i)^3\) is:

\[ x^3 - 9x + 27i \]