To simplify [tex]\(\sqrt{-144}\)[/tex], we need to recognize that the square root of a negative number involves the use of imaginary numbers.
1. First, separate the negative part from the positive part. We know that [tex]\( \sqrt{-144} = \sqrt{144 \cdot (-1)} \)[/tex].
2. The square root of a product is the product of the square roots, so we can write:
[tex]\[ \sqrt{144 \cdot (-1)} = \sqrt{144} \cdot \sqrt{-1} \][/tex]
3. The square root of 144 is 12 since [tex]\( 12 \cdot 12 = 144 \)[/tex]. Therefore:
[tex]\[ \sqrt{144} = 12 \][/tex]
4. The square root of [tex]\(-1\)[/tex] is defined as the imaginary unit [tex]\(i\)[/tex], which means:
[tex]\[ \sqrt{-1} = i \][/tex]
5. Combining these results:
[tex]\[ \sqrt{-144} = 12 \cdot i \][/tex]
Therefore, the simplification of [tex]\(\sqrt{-144}\)[/tex] is [tex]\(12i\)[/tex].
So, the correct answer is:
[tex]\[ 12i \][/tex]
