Use the imaginary number \( i \) to rewrite the expression below as a complex number. Simplify all radicals. \( -3+\sqrt{-64} \)

To rewrite the expression \( -3 + \sqrt{-64} \) as a complex number using the imaginary unit \( i \), follow these steps: 1. **Understand the Imaginary Unit \( i \):** The imaginary unit \( i \) is defined as: \[ i = \sqrt{-1} \] This means that any square root of a negative number can be expressed in terms of \( i \). 2. **Simplify the Radical \( \sqrt{-64} \):** Break down the expression inside the square root: \[ \sqrt{-64} = \sqrt{64 \times (-1)} \] Separate the square root of the product into the product of square roots: \[ \sqrt{64 \times (-1)} = \sqrt{64} \times \sqrt{-1} \] Simplify each part: \[ \sqrt{64} = 8 \quad \text{and} \quad \sqrt{-1} = i \] Therefore: \[ \sqrt{-64} = 8i \] 3. **Combine with the Real Part:** Now, substitute \( \sqrt{-64} \) back into the original expression: \[ -3 + \sqrt{-64} = -3 + 8i \] 4. **Final Complex Number:** The expression \( -3 + 8i \) is now in the standard form of a complex number, where: \[ \text{Real part} = -3 \quad \text{and} \quad \text{Imaginary part} = 8i \] **Answer:** \[ -3 + 8i \]