Use the imaginary number \( i \) to rewrite the expression below as a complex number. Simplify all radicals. \[ \sqrt{-9} \]

To rewrite \( \sqrt{-9} \) using the imaginary number \( i \), we start by recognizing that \( \sqrt{-1} = i \). Therefore, we can express the square root of a negative number as follows: \[ \sqrt{-9} = \sqrt{9} \cdot \sqrt{-1} = 3i \] Thus, the expression simplifies to the complex number \( 3i \). Now for some fun historical context: The concept of imaginary numbers dates back to the 16th century when mathematicians were trying to solve equations that didn't have real solutions, particularly in cases like the square root of negative numbers. It wasn't until the 18th century that mathematicians like Euler began to formalize and embrace these "imaginary" numbers, leading to the rich field of complex analysis we study today! And what about some practical advice? When dealing with complex numbers, always remember to separate the real part from the imaginary part, especially when you're adding or subtracting different complex numbers. A common mistake is to forget that when multiplying, you should distribute carefully considering that \( i^2 = -1 \). This often trips people up when simplifying products of complex numbers!