To solve the expression [tex]\(\sqrt{-6} \cdot \sqrt{-24}\)[/tex], we will use the concept of imaginary numbers.
Given that [tex]\( i = \sqrt{-1} \)[/tex], we can rewrite the square roots of negative numbers using [tex]\(i\)[/tex].
1. First, consider [tex]\(\sqrt{-6}\)[/tex]:
[tex]\[
\sqrt{-6} = \sqrt{6} \cdot \sqrt{-1} = \sqrt{6} \cdot i
\][/tex]
Thus, we can write:
[tex]\[
\sqrt{-6} = \sqrt{6} \cdot i
\][/tex]
2. Next, consider [tex]\(\sqrt{-24}\)[/tex]:
[tex]\[
\sqrt{-24} = \sqrt{24} \cdot \sqrt{-1} = \sqrt{24} \cdot i
\][/tex]
Thus, we can write:
[tex]\[
\sqrt{-24} = \sqrt{24} \cdot i
\][/tex]
3. Now we multiply these two results together:
[tex]\[
(\sqrt{6} \cdot i) \cdot (\sqrt{24} \cdot i)
\][/tex]
4. Simplify by multiplying the square roots and [tex]\(i\)[/tex]'s separately:
[tex]\[
\sqrt{6} \cdot \sqrt{24} \cdot i \cdot i
\][/tex]
5. Recall that [tex]\(i \cdot i = i^2\)[/tex] and by definition, [tex]\(i^2 = -1\)[/tex]:
[tex]\[
\sqrt{6} \cdot \sqrt{24} \cdot (-1)
\][/tex]
6. Simplify the product of the square roots:
[tex]\[
\sqrt{6 \cdot 24} \cdot (-1)
\][/tex]
[tex]\[
\sqrt{144} \cdot (-1)
\][/tex]
7. Recognize that [tex]\(\sqrt{144} = 12\)[/tex]:
[tex]\[
12 \cdot (-1)
\][/tex]
8. Thus, we have:
[tex]\[
12 \cdot (-1) = -12
\][/tex]
Hence, the expression [tex]\(\sqrt{-6} \cdot \sqrt{-24}\)[/tex] simplifies to [tex]\(-12\)[/tex].
So the final result is:
[tex]\(\boxed{-12}\)[/tex]
