To solve for the sum of [tex]\(\sqrt{-2}\)[/tex] and [tex]\(\sqrt{-18}\)[/tex], let's proceed step-by-step.
First, we recall that both [tex]\(\sqrt{-2}\)[/tex] and [tex]\(\sqrt{-18}\)[/tex] involve the square roots of negative numbers, which means they will result in complex numbers due to the presence of the imaginary unit [tex]\(i\)[/tex] (where [tex]\(i = \sqrt{-1}\)[/tex]).
1. Calculate [tex]\(\sqrt{-2}\)[/tex]:
The square root of [tex]\(-2\)[/tex] can be expressed in terms of [tex]\(i\)[/tex]:
[tex]\[
\sqrt{-2} = \sqrt{2} \cdot \sqrt{-1} = \sqrt{2} \cdot i
\][/tex]
So,
[tex]\[
\sqrt{-2} = \sqrt{2} \cdot i
\][/tex]
Numerically, [tex]\(\sqrt{2} \approx 1.4142135623730951\)[/tex].
2. Calculate [tex]\(\sqrt{-18}\)[/tex]:
Similarly, the square root of [tex]\(-18\)[/tex] can be simplified as:
[tex]\[
\sqrt{-18} = \sqrt{18} \cdot \sqrt{-1} = \sqrt{18} \cdot i = \sqrt{9 \cdot 2} \cdot i = \sqrt{9} \cdot \sqrt{2} \cdot i = 3 \sqrt{2} \cdot i
\][/tex]
Since [tex]\(\sqrt{9}=3\)[/tex], we have:
[tex]\[
\sqrt{-18} = 3 \cdot \sqrt{2} \cdot i
\][/tex]
Numerically, [tex]\(3 \cdot \sqrt{2} \approx 4.242640687119285\)[/tex].
3. Sum the results:
Now, let's add [tex]\(\sqrt{-2}\)[/tex] and [tex]\(\sqrt{-18}\)[/tex]:
[tex]\[
\sqrt{-2} + \sqrt{-18} = \sqrt{2} \cdot i + 3 \cdot \sqrt{2} \cdot i
\][/tex]
Factoring out [tex]\(\sqrt{2} \cdot i\)[/tex] gives:
[tex]\[
\sqrt{2} \cdot i + 3 \cdot \sqrt{2} \cdot i = (1 + 3) \cdot \sqrt{2} \cdot i = 4 \cdot \sqrt{2} \cdot i
\][/tex]
Numerically, [tex]\(4 \cdot \sqrt{2} \cdot i \approx 5.65685424949238i\)[/tex].
Therefore, the sum of [tex]\(\sqrt{-2}\)[/tex] and [tex]\(\sqrt{-18}\)[/tex] is:
[tex]\[
\boxed{4 \cdot i \cdot \sqrt{2}}
\][/tex]
So, the correct answer from the given choices is [tex]\(4 i \sqrt{2}\)[/tex].
