Use the imaginary number i to rewrite the expression below as a complex number, Simplify all radicals. \( 12+\sqrt{-16} \)

To rewrite the expression \( 12 + \sqrt{-16} \) as a complex number using the imaginary unit \( i \), follow these steps: 1. **Understand the Imaginary Unit**: - The imaginary unit \( i \) is defined by the property \( i^2 = -1 \). - This allows us to express the square root of a negative number in terms of \( i \). 2. **Simplify the Radical**: - Start with the expression \( \sqrt{-16} \). - Break it down using the property of square roots: \( \sqrt{-16} = \sqrt{16 \times (-1)} \). - Separate the square roots: \( \sqrt{16} \times \sqrt{-1} \). - Compute each part: \( \sqrt{16} = 4 \) and \( \sqrt{-1} = i \). - Therefore, \( \sqrt{-16} = 4i \). 3. **Combine with the Real Part**: - Now, substitute \( \sqrt{-16} \) back into the original expression: \[ 12 + \sqrt{-16} = 12 + 4i \] 4. **Final Answer**: - The expression is now written as a complex number in the form \( a + bi \), where \( a \) and \( b \) are real numbers. \[ 12 + 4i \]