For `i^2= -1, (1 - 3i)^3= ?`

Options

A. `-26`
B. `-8`
C. `-10+30i`
D. `-26+18i`

Answer

To solve the expression \((1 - 3i)^3\), we can use the binomial expansion formula for the cube of a binomial, which is given by:

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

In our case, let \(a = 1\) and \(b = 3i\). Now, we can substitute these values into the formula.

### Step 1: Calculate \(a^3\)
\[ a^3 = 1^3 = 1 \]

### Step 2: Calculate \(-3a^2b\)
\[ -3a^2b = -3(1^2)(3i) = -3(1)(3i) = -9i \]

### Step 3: Calculate \(3ab^2\)
\[ b^2 = (3i)^2 = 9i^2 \]
Since \(i^2 = -1\),
\[ b^2 = 9(-1) = -9 \]
Now, substituting back:
\[ 3ab^2 = 3(1)(-9) = -27 \]

### Step 4: Calculate \(-b^3\)
\[ -b^3 = -(3i)^3 = -27i^3 \]
Since \(i^3 = i^2 \cdot i = -1 \cdot i = -i\),
\[ -b^3 = -27(-i) = 27i \]

### Step 5: Combine all the terms
Now we combine all the calculated terms:
\[ (1 - 3i)^3 = 1 - 9i - 27 + 27i \]

### Step 6: Simplify the expression
Combining like terms:
\[ 1 - 27 = -26 \]
\[ -9i + 27i = 18i \]
Thus, we have:
\[ (1 - 3i)^3 = -26 + 18i \]

### Final Answer
The final result is:
\[ (1 - 3i)^3 = -26 + 18i \]

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