`(-2-(1)/(3)i)^(3)`

Explanation

To evaluate the expression \((-2 - \frac{1}{3}i)^3\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ (-2 - \frac{1}{3}i)^3 \]

### Step 2: Factor out \(-1\)
We can factor out \(-1\) from the expression:
\[ = (-1)^3 \cdot (2 + \frac{1}{3}i)^3 \]
This simplifies to:
\[ = - (2 + \frac{1}{3}i)^3 \]

### Step 3: Use the binomial expansion
Now we apply the binomial expansion formula \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\), where \(a = 2\) and \(b = \frac{1}{3}i\):
\[ = - \left(2^3 + 3 \cdot 2^2 \cdot \frac{1}{3}i + 3 \cdot 2 \cdot \left(\frac{1}{3}i\right)^2 + \left(\frac{1}{3}i\right)^3\right) \]

### Step 4: Calculate each term
Calculating each term:
- \(2^3 = 8\)
- \(3 \cdot 2^2 \cdot \frac{1}{3}i = 3 \cdot 4 \cdot \frac{1}{3}i = 4i\)
- \(3 \cdot 2 \cdot \left(\frac{1}{3}i\right)^2 = 3 \cdot 2 \cdot \frac{-1}{9} = -\frac{6}{9} = -\frac{2}{3}\)
- \(\left(\frac{1}{3}i\right)^3 = \frac{1}{27}i^3 = \frac{1}{27}(-i) = -\frac{1}{27}i\)

### Step 5: Combine the terms
Now we combine all these terms:
\[ = - \left(8 + 4i - \frac{2}{3} - \frac{1}{27}i\right) \]
Combining the real parts:
\[ = - \left(8 - \frac{2}{3}\right) + - \left(4i - \frac{1}{27}i\right) \]

### Step 6: Simplify the real part
To combine \(8\) and \(-\frac{2}{3}\):
\[ 8 = \frac{24}{3} \quad \Rightarrow \quad \frac{24}{3} - \frac{2}{3} = \frac{22}{3} \]
So the real part becomes:
\[ - \frac{22}{3} \]

### Step 7: Simplify the imaginary part
Now for the imaginary part:
\[ 4i - \frac{1}{27}i = \left(4 - \frac{1}{27}\right)i \]
To combine \(4\) and \(-\frac{1}{27}\):
\[ 4 = \frac{108}{27} \quad \Rightarrow \quad \frac{108}{27} - \frac{1}{27} = \frac{107}{27} \]
So the imaginary part becomes:
\[ -\frac{107}{27}i \]

### Step 8: Final expression
Putting it all together, we have:
\[ = -\left(\frac{22}{3} + \frac{107}{27}i\right) \]
Thus, the final result is:
\[ = -\frac{22}{3} - \frac{107}{27}i \]