Evaluate: `sqrt(-15-8i)`

Answer

To evaluate \( \sqrt{-15 - 8i} \), we can express the square root in terms of real and imaginary parts. Let's denote:

\[ \sqrt{-15 - 8i} = x + iy \]

where \( x \) and \( y \) are real numbers.

### Step 1: Square both sides
Squaring both sides gives us:

\[ x^2 + 2xyi - y^2 = -15 - 8i \]

This means we can separate the real and imaginary parts:

\[ x^2 - y^2 = -15 \quad \text{(1)} \]
\[ 2xy = -8 \quad \text{(2)} \]

### Step 2: Solve for \( y \)
From equation (2), we can express \( y \) in terms of \( x \):

\[ y = \frac{-8}{2x} = \frac{-4}{x} \]

### Step 3: Substitute \( y \) into equation (1)
Now, substitute \( y \) into equation (1):

\[ x^2 - \left(\frac{-4}{x}\right)^2 = -15 \]

This simplifies to:

\[ x^2 - \frac{16}{x^2} = -15 \]

### Step 4: Multiply through by \( x^2 \)
To eliminate the fraction, multiply through by \( x^2 \):

\[ x^4 + 15x^2 - 16 = 0 \]

### Step 5: Let \( t = x^2 \)
Let \( t = x^2 \). The equation becomes:

\[ t^2 + 15t - 16 = 0 \]

### Step 6: Solve the quadratic equation
We can solve this quadratic equation using the quadratic formula:

\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

where \( a = 1, b = 15, c = -16 \):

\[ t = \frac{-15 \pm \sqrt{15^2 - 4 \cdot 1 \cdot (-16)}}{2 \cdot 1} \]
\[ t = \frac{-15 \pm \sqrt{225 + 64}}{2} \]
\[ t = \frac{-15 \pm \sqrt{289}}{2} \]
\[ t = \frac{-15 \pm 17}{2} \]

Calculating the two possible values for \( t \):

1. \( t = \frac{2}{2} = 1 \)
2. \( t = \frac{-32}{2} = -16 \) (not valid since \( t = x^2 \) must be non-negative)

Thus, we have:

\[ x^2 = 1 \implies x = 1 \text{ or } x = -1 \]

### Step 7: Find \( y \)
Now substituting back to find \( y \):

1. If \( x = 1 \):
\[ y = \frac{-4}{1} = -4 \]

2. If \( x = -1 \):
\[ y = \frac{-4}{-1} = 4 \]

### Step 8: Write the final results
Thus, we have two possible solutions for \( \sqrt{-15 - 8i} \):

\[ \sqrt{-15 - 8i} = 1 - 4i \quad \text{or} \quad -1 + 4i \]

### Final Answer
The values of \( \sqrt{-15 - 8i} \) are:

\[ 1 - 4i \quad \text{and} \quad -1 + 4i \]