If `(x + iy) ^( 1//3) = 2 + 3i,` then `3x + 2y ` is equal to

Options

A. `- 20`
B. `- 60`
C. `- 120`
D. `60`

Answer

To solve the problem, we start with the equation given:

\[ (x + iy)^{\frac{1}{3}} = 2 + 3i \]

### Step 1: Cube both sides
To eliminate the cube root, we will cube both sides of the equation:

\[ x + iy = (2 + 3i)^3 \]

### Step 2: Calculate \((2 + 3i)^3\)
We will now calculate \((2 + 3i)^3\) using the binomial theorem or by direct multiplication:

\[ (2 + 3i)^3 = (2 + 3i)(2 + 3i)(2 + 3i) \]

First, calculate \((2 + 3i)(2 + 3i)\):

\[ (2 + 3i)(2 + 3i) = 4 + 12i + 9i^2 = 4 + 12i - 9 = -5 + 12i \]

Now, multiply \((-5 + 12i)(2 + 3i)\):

\[ (-5 + 12i)(2 + 3i) = -10 - 15i + 24i + 36i^2 = -10 + 9i - 36 = -46 + 9i \]

So, we have:

\[ x + iy = -46 + 9i \]

### Step 3: Equate real and imaginary parts
From the equation \(x + iy = -46 + 9i\), we can equate the real and imaginary parts:

\[ x = -46 \quad \text{and} \quad y = 9 \]

### Step 4: Calculate \(3x + 2y\)
Now we need to find \(3x + 2y\):

\[ 3x + 2y = 3(-46) + 2(9) \]

Calculating this gives:

\[ 3(-46) = -138 \quad \text{and} \quad 2(9) = 18 \]

Now combine these results:

\[ 3x + 2y = -138 + 18 = -120 \]

### Final Answer
Thus, the value of \(3x + 2y\) is:

\[ \boxed{-120} \]