Use the imaginary number i to rewrite the expression below as a complex number. Simplify all radicals. \[ 1+\sqrt{-63} \]

To rewrite the expression \( 1 + \sqrt{-63} \) as a complex number using the imaginary unit \( i \), follow these steps: 1. **Express the Square Root of a Negative Number:** \[ \sqrt{-63} = \sqrt{63} \cdot i \] Here, \( i \) is the imaginary unit, defined by \( i^2 = -1 \). 2. **Simplify the Radical \( \sqrt{63} \):** \[ \sqrt{63} = \sqrt{9 \times 7} = \sqrt{9} \times \sqrt{7} = 3\sqrt{7} \] 3. **Combine the Results:** \[ \sqrt{-63} = 3\sqrt{7} \cdot i \] 4. **Write the Complete Complex Number:** \[ 1 + \sqrt{-63} = 1 + 3\sqrt{7}i \] **Final Answer:** \[ 1 + 3\,\sqrt{7}\,i \]