Certainly! Let's solve the problem step-by-step:
1. Calculate the square root of [tex]\(-2\)[/tex]:
- To find [tex]\(\sqrt{-2}\)[/tex], we remember that [tex]\(\sqrt{-a} = \sqrt{a} \cdot i\)[/tex], where [tex]\(i\)[/tex] is the imaginary unit ([tex]\(i^2 = -1\)[/tex]).
- Therefore, [tex]\(\sqrt{-2} = \sqrt{2} \cdot i\)[/tex].
2. Calculate the square root of [tex]\(-18\)[/tex]:
- Similarly, for [tex]\(\sqrt{-18}\)[/tex], we apply the same principle:
- First, [tex]\(\sqrt{18}\)[/tex] can be broken down into [tex]\(\sqrt{2 \cdot 9} = \sqrt{2} \cdot \sqrt{9} = 3 \sqrt{2}\)[/tex].
- Therefore, [tex]\(\sqrt{-18} = \sqrt{18} \cdot i = 3 \sqrt{2} \cdot i\)[/tex].
3. Add the two results together:
- We now have [tex]\(\sqrt{-2} = \sqrt{2} \cdot i\)[/tex] and [tex]\(\sqrt{-18} = 3 \sqrt{2} \cdot i\)[/tex].
- Adding these, we get:
[tex]\[
\sqrt{-2} + \sqrt{-18} = \sqrt{2} \cdot i + 3 \sqrt{2} \cdot i
\][/tex]
- Factor out the common term [tex]\(\sqrt{2} \cdot i\)[/tex]:
[tex]\[
\sqrt{2} \cdot i (1 + 3) = \sqrt{2} \cdot i \cdot 4 = 4 \sqrt{2} \cdot i
\][/tex]
4. Final Result:
- The sum of [tex]\(\sqrt{-2}\)[/tex] and [tex]\(\sqrt{-18}\)[/tex] is [tex]\(4 \sqrt{2} \cdot i\)[/tex].
Therefore, the correct answer is [tex]\(4i \sqrt{2}\)[/tex].
