To find the sum of [tex]\(\sqrt{-2}\)[/tex] and [tex]\(\sqrt{-18}\)[/tex], we need to deal with the square roots of negative numbers, which involve the imaginary unit [tex]\(i\)[/tex].
1. Identify the square roots with imaginary units:
- [tex]\(\sqrt{-2}\)[/tex] can be rewritten as [tex]\(\sqrt{2} \cdot i\)[/tex].
- [tex]\(\sqrt{-18}\)[/tex] can be rewritten as [tex]\(\sqrt{18} \cdot i\)[/tex].
2. Simplify the square roots:
- [tex]\(\sqrt{2}\)[/tex] remains [tex]\(\sqrt{2}\)[/tex].
- [tex]\(\sqrt{18}\)[/tex] can be simplified as follows:
[tex]\[
\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \cdot \sqrt{2} = 3 \cdot \sqrt{2}
\][/tex]
Therefore, [tex]\(\sqrt{-18} = 3 \cdot \sqrt{2} \cdot i\)[/tex].
3. Rewrite the terms in terms of the imaginary unit [tex]\(i\)[/tex]:
- [tex]\(\sqrt{-2} = \sqrt{2} \cdot i\)[/tex]
- [tex]\(\sqrt{-18} = 3 \cdot \sqrt{2} \cdot i\)[/tex]
4. Sum the imaginary parts:
[tex]\[
\sqrt{2} \cdot i + 3 \cdot \sqrt{2} \cdot i
\][/tex]
Combine the like terms:
[tex]\[
(\sqrt{2} + 3 \cdot \sqrt{2}) \cdot i = 4 \cdot \sqrt{2} \cdot i
\][/tex]
Therefore, the sum of [tex]\(\sqrt{-2}\)[/tex] and [tex]\(\sqrt{-18}\)[/tex] is [tex]\(4i \sqrt{2}\)[/tex]. This corresponds to option:
[tex]\[ \boxed{4i \sqrt{2}} \][/tex]
