Simplify the following number by using the imaginary number \( i \). \[ \sqrt{-150} \]

To simplify \( \sqrt{-150} \), we can start by expressing the negative number using the imaginary unit \( i \). We know that \( \sqrt{-1} = i \). Thus, \[ \sqrt{-150} = \sqrt{150} \cdot \sqrt{-1} = \sqrt{150} i \] Next, we can simplify \( \sqrt{150} \). The number 150 can be factored into \( 25 \times 6 \), where 25 is a perfect square: \[ \sqrt{150} = \sqrt{25 \times 6} = \sqrt{25} \cdot \sqrt{6} = 5\sqrt{6} \] Putting it all together, we have: \[ \sqrt{-150} = 5\sqrt{6} i \] So, the simplified form of \( \sqrt{-150} \) is \( 5\sqrt{6} i \).