Use the imaginary number \( i \) to rewrite the expression below as a complex number. Simplify all radicals. \( 8+\sqrt{-96} \) \begin{tabular}{|l}\hline\( i \)\end{tabular}

To rewrite the expression \( 8 + \sqrt{-96} \) as a complex number, we can first simplify the square root of a negative number by recognizing that \( \sqrt{-1} = i \). Thus, we have: \[ \sqrt{-96} = \sqrt{96} \cdot \sqrt{-1} = \sqrt{96}i \] Next, we simplify \( \sqrt{96} \). The number 96 can be factored as: \[ 96 = 16 \times 6 = 4^2 \times 6 \] So, \[ \sqrt{96} = \sqrt{16 \times 6} = \sqrt{16} \cdot \sqrt{6} = 4\sqrt{6} \] Now we replace \( \sqrt{96} \) back into the expression: \[ \sqrt{-96} = 4\sqrt{6}i \] Substituting this back into the original expression gives: \[ 8 + \sqrt{-96} = 8 + 4\sqrt{6}i \] Therefore, the simplified form of the expression as a complex number is: \[ 8 + 4\sqrt{6}i \]