If ` 27 i z^(3) + 18 z^(2) - 12 z + 8i = 0, ` then |z| =

Options

A. `(2)/(3)`
B. `(4)/(9)`
C. 1
D. none

Answer

To solve the equation \( 27i z^3 + 18z^2 - 12z + 8i = 0 \) and find the modulus of \( z \), we can follow these steps:

### Step 1: Factor out common terms
We start by factoring out \( 9z^2 \) from the first two terms:

\[ 27i z^3 + 18z^2 - 12z + 8i = 0 \]

This can be rewritten as:

\[ 9z^2(3iz + 2) - 12z + 8i = 0 \]

### Step 2: Rearranging the equation
Next, we rearrange the equation to isolate the terms involving \( z \):

\[ 9z^2(3iz + 2) = 12z - 8i \]

### Step 3: Set the equation to zero
We can set the equation to zero by moving all terms to one side:

\[ 9z^2(3iz + 2) - 12z + 8i = 0 \]

### Step 4: Identify factors
Notice that we can express this as:

\[ (3iz + 2)(9z^2 - 4) = 0 \]

### Step 5: Solve for \( z \)
Now, we can set each factor to zero.

1. From \( 3iz + 2 = 0 \):
\[ 3iz = -2 \implies z = -\frac{2}{3i} = \frac{2i}{3} \]

2. From \( 9z^2 - 4 = 0 \):
\[ 9z^2 = 4 \implies z^2 = \frac{4}{9} \implies z = \pm \frac{2}{3} \]

### Step 6: Find the modulus of \( z \)
Now, we need to find the modulus of \( z \):
- For \( z = \frac{2i}{3} \):
\[ |z| = \left| \frac{2i}{3} \right| = \frac{2}{3} \]

- For \( z = \frac{2}{3} \):
\[ |z| = \left| \frac{2}{3} \right| = \frac{2}{3} \]

Thus, the modulus of \( z \) is:

\[ |z| = \frac{2}{3} \]

### Final Answer:
\[ |z| = \frac{2}{3} \]

---