The value of `(3sqrt(3)+(3^(5//6))i)^(3)` is (where `i=sqrt(-1)`)

Options

A. `24`
B. `-24`
C. `-22`
D. `-21`

Explanation

To solve the expression \((3\sqrt{3} + 3^{5/6}i)^3\), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ (3\sqrt{3} + 3^{5/6}i)^3 \]
We can express \(3\sqrt{3}\) in terms of powers of 3:
\[ 3\sqrt{3} = 3^{1} \cdot 3^{1/2} = 3^{3/2} \]
Thus, we can rewrite the expression as:
\[ (3^{3/2} + 3^{5/6}i)^3 \]

### Step 2: Factor out a common term
Next, we can factor out \(3^{5/6}\) from both terms:
\[ = \left(3^{5/6}(3^{3/2 - 5/6} + i)\right)^3 \]
Calculating \(3^{3/2 - 5/6}\):
\[ 3^{3/2} = 3^{9/6} \quad \text{so} \quad 3^{3/2 - 5/6} = 3^{9/6 - 5/6} = 3^{4/6} = 3^{2/3} \]
Now we have:
\[ = \left(3^{5/6}(3^{2/3} + i)\right)^3 \]

### Step 3: Expand the expression
Using the property of exponents:
\[ = (3^{5/6})^3 \cdot (3^{2/3} + i)^3 \]
Calculating \((3^{5/6})^3\):
\[ (3^{5/6})^3 = 3^{15/6} = 3^{5/2} \]
So now we have:
\[ = 3^{5/2} \cdot (3^{2/3} + i)^3 \]

### Step 4: Calculate \((3^{2/3} + i)^3\)
We can use the binomial expansion:
\[ (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \]
Let \(a = 3^{2/3}\) and \(b = i\):
\[ = (3^{2/3})^3 + 3(3^{2/3})^2(i) + 3(3^{2/3})(i^2) + i^3 \]
Calculating each term:
1. \((3^{2/3})^3 = 3^{2}\)
2. \(3(3^{2/3})^2(i) = 3 \cdot 3^{4/3}i = 3^{7/3}i\)
3. \(3(3^{2/3})(i^2) = 3(3^{2/3})(-1) = -3^{5/3}\)
4. \(i^3 = -i\)

Combining these:
\[ = 3^2 + 3^{7/3}i - 3^{5/3} - i \]
\[ = 9 - 3^{5/3} + (3^{7/3} - 1)i \]

### Step 5: Combine the results
Now substituting back:
\[ = 3^{5/2} \cdot (9 - 3^{5/3} + (3^{7/3} - 1)i) \]
Distributing \(3^{5/2}\):
\[ = 3^{5/2} \cdot 9 - 3^{5/2} \cdot 3^{5/3} + 3^{5/2} \cdot (3^{7/3} - 1)i \]
Calculating \(3^{5/2} \cdot 3^{5/3}\):
\[ 3^{5/2 + 5/3} = 3^{(15/6 + 10/6)} = 3^{25/6} \]
Calculating \(3^{5/2} \cdot (3^{7/3} - 1)\):
\[ 3^{5/2} \cdot 3^{7/3} = 3^{(15/6 + 14/6)} = 3^{29/6} \]
Thus, we have:
\[ = 3^{5/2} \cdot 9 - 3^{25/6} + (3^{29/6} - 3^{5/2})i \]

### Final Answer
The value of \((3\sqrt{3} + 3^{5/6}i)^3\) is:
\[ = 3^{5/2} \cdot 9 - 3^{25/6} + (3^{29/6} - 3^{5/2})i \]