To find the sum of [tex]\(\sqrt{-2}\)[/tex] and [tex]\(\sqrt{-18}\)[/tex], we need to first express these square roots in terms of imaginary numbers.
1. Calculate [tex]\(\sqrt{-2}\)[/tex]:
[tex]\[
\sqrt{-2} = \sqrt{2 \cdot -1} = \sqrt{2} \cdot \sqrt{-1} = \sqrt{2} \cdot i = i \sqrt{2}
\][/tex]
2. Calculate [tex]\(\sqrt{-18}\)[/tex]:
[tex]\[
\sqrt{-18} = \sqrt{18 \cdot -1} = \sqrt{18} \cdot \sqrt{-1} = \sqrt{18} \cdot i
\][/tex]
Next, simplify [tex]\(\sqrt{18}\)[/tex]:
[tex]\[
\sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3 \cdot \sqrt{2} = 3\sqrt{2}
\][/tex]
Thus,
[tex]\[
\sqrt{-18} = 3\sqrt{2} \cdot i = 3i \sqrt{2}
\][/tex]
3. Sum up [tex]\(\sqrt{-2}\)[/tex] and [tex]\(\sqrt{-18}\)[/tex]:
Add [tex]\(i \sqrt{2}\)[/tex] and [tex]\(3i \sqrt{2}\)[/tex]:
[tex]\[
i \sqrt{2} + 3i \sqrt{2} = (1i \sqrt{2} + 3i \sqrt{2}) = 4i \sqrt{2}
\][/tex]
Therefore, the sum of [tex]\(\sqrt{-2}\)[/tex] and [tex]\(\sqrt{-18}\)[/tex] is:
[tex]\[
4i \sqrt{2}
\][/tex]
So the correct answer is:
[tex]\[
\boxed{4 i \sqrt{2}}
\][/tex]
