Use the imaginary number \( i \) to rewrite the expression below as a complex number. Simplify all radicals. \( \sqrt{-74} \) \begin{tabular}{|c|c}\hline \\ \hline\end{tabular}\( \sqrt{ } \)

To rewrite \( \sqrt{-74} \) as a complex number using the imaginary unit \( i \), follow these steps: 1. **Identify the Negative Sign:** The square root of a negative number can be expressed using \( i \), where \( i = \sqrt{-1} \). 2. **Separate the Radicand:** \[ \sqrt{-74} = \sqrt{74 \times -1} = \sqrt{74} \times \sqrt{-1} \] 3. **Apply the Definition of \( i \):** \[ \sqrt{-1} = i \] 4. **Combine the Terms:** \[ \sqrt{-74} = \sqrt{74} \times i = \sqrt{74}\,i \] 5. **Simplify the Radical (if possible):** - Check if 74 can be factored into a perfect square and another factor. - \( 74 = 2 \times 37 \) - Neither 2 nor 37 are perfect squares, so \( \sqrt{74} \) cannot be simplified further. **Final Answer:** \[ \sqrt{-74} = \sqrt{74}\,i \]