If z is a complex number satisfying the equaiton `z^(6) - 6z^(3) + 25 = 0`, then the value of `|z|` is

Options

A. `5^(1//3)`
B. `25^(1//3)`
C. `125^(1//3)`
D. `625^(1//3)`

Explanation

To solve the equation \( z^6 - 6z^3 + 25 = 0 \) for the value of \( |z| \), we can follow these steps:

### Step 1: Substitute \( z^3 \) with \( t \)
Let \( t = z^3 \). Then, we can rewrite the equation as:
\[ t^2 - 6t + 25 = 0 \]

### Step 2: Apply the quadratic formula
The quadratic formula is given by:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our case, \( a = 1 \), \( b = -6 \), and \( c = 25 \). Plugging in these values:
\[ t = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot 25}}{2 \cdot 1} \]
This simplifies to:
\[ t = \frac{6 \pm \sqrt{36 - 100}}{2} \]
\[ t = \frac{6 \pm \sqrt{-64}}{2} \]

### Step 3: Simplify the square root
Since \( \sqrt{-64} = 8i \), we can write:
\[ t = \frac{6 \pm 8i}{2} \]
This gives us:
\[ t = 3 \pm 4i \]

### Step 4: Find \( |z^3| \)
Now, we have two possible values for \( t \):
1. \( t = 3 + 4i \)
2. \( t = 3 - 4i \)

To find \( |z| \), we first need to find \( |z^3| \):
\[ |t| = |z^3| = |3 + 4i| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 5: Find \( |z| \)
Since \( |z^3| = |z|^3 \), we have:
\[ |z|^3 = 5 \]
Taking the cube root of both sides, we find:
\[ |z| = 5^{1/3} \]

### Final Answer
Thus, the value of \( |z| \) is:
\[ |z| = \sqrt[3]{5} \]