To solve the problem [tex]\(\sqrt{-9}\)[/tex], we need to understand that finding the square root of a negative number results in an imaginary number.
The imaginary unit, denoted by [tex]\(i\)[/tex], is defined as [tex]\(i = \sqrt{-1}\)[/tex]. Using this, we can rewrite negative numbers under the square root as follows:
1. Acknowledge that [tex]\(\sqrt{-9}\)[/tex] can be written as [tex]\(\sqrt{9 \times -1}\)[/tex].
2. This can be further broken down using the properties of square roots: [tex]\(\sqrt{9 \times -1} = \sqrt{9} \cdot \sqrt{-1}\)[/tex].
3. Since [tex]\(\sqrt{-1} = i\)[/tex], we substitute to get: [tex]\(\sqrt{9} \cdot i\)[/tex].
4. Then, we know that [tex]\(\sqrt{9} = 3\)[/tex], which gives us: [tex]\( 3 \cdot i\)[/tex].
Hence, [tex]\(\sqrt{-9} = 3i\)[/tex].
So the answer is:
[tex]\[
\sqrt{-9} = 3i
\][/tex]
