Let `z` be a complex number satisfying the equation `(z^3+3)^2=-16` , then find the value of `|z|`

Explanation

To solve the equation \((z^3 + 3)^2 = -16\) and find the value of \(|z|\), we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ (z^3 + 3)^2 = -16 \]
We can express \(-16\) in terms of \(i\) (the imaginary unit):
\[ -16 = 16(-1) = 16i^2 \]
Thus, we can rewrite the equation as:
\[ (z^3 + 3)^2 = 16i^2 \]

### Step 2: Take the square root
Taking the square root of both sides, we have:
\[ z^3 + 3 = 4i \quad \text{or} \quad z^3 + 3 = -4i \]

### Step 3: Solve for \(z^3\)
From the first case:
\[ z^3 = 4i - 3 \]
From the second case:
\[ z^3 = -4i - 3 \]

### Step 4: Calculate the modulus of \(z^3\)
Now we will calculate the modulus for both cases.

**Case 1:**
\[ z^3 = 4i - 3 \]
To find the modulus:
\[ |z^3| = |4i - 3| = \sqrt{(-3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

**Case 2:**
\[ z^3 = -4i - 3 \]
To find the modulus:
\[ |z^3| = |-4i - 3| = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

In both cases, we find:
\[ |z^3| = 5 \]

### Step 5: Relate the modulus of \(z\) to the modulus of \(z^3\)
Using the property of moduli:
\[ |z^3| = |z|^3 \]
Thus:
\[ |z|^3 = 5 \]

### Step 6: Solve for \(|z|\)
To find \(|z|\):
\[ |z| = \sqrt[3]{5} \]

### Final Answer
The value of \(|z|\) is:
\[ |z| = 5^{1/3} \]