To simplify the expression [tex]\(-5 - \sqrt{-44}\)[/tex], we need to break it down step-by-step.
1. Understanding the Square Root of a Negative Number:
The square root of a negative number involves the imaginary unit [tex]\(i\)[/tex], where [tex]\(i = \sqrt{-1}\)[/tex].
2. Rewrite the Square Root:
The given expression involves [tex]\(\sqrt{-44}\)[/tex]:
[tex]\[
\sqrt{-44} = \sqrt{-1 \cdot 44} = \sqrt{-1} \cdot \sqrt{44} = i \cdot \sqrt{44}
\][/tex]
3. Calculate [tex]\(\sqrt{44}\)[/tex]:
The square root of 44 can be evaluated as:
[tex]\[
\sqrt{44} \approx 6.6332495807108
\][/tex]
4. Combine with the Imaginary Unit:
Using the imaginary unit [tex]\(i\)[/tex]:
[tex]\[
i \cdot \sqrt{44} \approx 6.6332495807108i
\][/tex]
5. Substitute Back into the Original Expression:
Now substitute [tex]\(i \cdot \sqrt{44}\)[/tex] back into the original expression:
[tex]\[
-5 - \sqrt{-44} = -5 - 6.6332495807108i
\][/tex]
6. Formulate the Final Simplified Expression:
When we look at the options provided:
- [tex]\(-5 - 4 \sqrt{11i}\)[/tex]
- [tex]\(-5 - 4i \sqrt{11}\)[/tex]
- [tex]\(-5 - 2i \sqrt{11}\)[/tex]
- [tex]\(-5 - 2 \sqrt{11i}\)[/tex]
The expression [tex]\(-5 -2i \sqrt{11}\)[/tex] looks correct because it simplifies to:
[tex]\[
\sqrt{44} = \sqrt{4 \cdot 11} = 2\sqrt{11}
\][/tex]
So,
[tex]\[
\sqrt{-44} = i \cdot 2\sqrt{11}
= 2i\sqrt{11}
\][/tex]
Hence:
[tex]\[
-5 - \sqrt{-44} = -5 - 2i\sqrt{11}
\][/tex]
This matches the third option.
Therefore, the correct simplified form is:
[tex]\[
\boxed{-5 - 2i \sqrt{11}}
\][/tex]
And numerically, this matches to:
[tex]\[
(-5 - 13.2664991614216i)
\][/tex]
