Simplify the imaginary number. \( 2 \sqrt{-112} \)

To simplify \( 2 \sqrt{-112} \), first, we can rewrite the expression as \( 2 \sqrt{112} \cdot \sqrt{-1} \). Since \( \sqrt{-1} \) is denoted as \( i \), we have: \[ 2 \sqrt{-112} = 2 \sqrt{112} i \] Now, simplify \( \sqrt{112} \). We can factor it: \[ \sqrt{112} = \sqrt{16 \cdot 7} = \sqrt{16} \cdot \sqrt{7} = 4\sqrt{7} \] Now substituting back, we get: \[ 2 \sqrt{-112} = 2 \cdot 4\sqrt{7} i = 8\sqrt{7} i \] So, the simplified form of the imaginary number is: \[ 8\sqrt{7} i \]