To find the value of [tex]\(-5 - \sqrt{-44}\)[/tex], we need to simplify the expression involving the square root of a negative number. Here's a detailed step-by-step solution:
1. Expressing the square root of a negative number:
[tex]\[
\sqrt{-44} = \sqrt{-1 \cdot 44}
\][/tex]
2. Breaking it down using the property of square roots:
[tex]\[
\sqrt{-44} = \sqrt{-1} \cdot \sqrt{44}
\][/tex]
3. Understanding [tex]\(\sqrt{-1}\)[/tex]:
By definition, [tex]\(\sqrt{-1}\)[/tex] is represented by the imaginary unit [tex]\(i\)[/tex].
[tex]\[
\sqrt{-44} = i \cdot \sqrt{44}
\][/tex]
4. Simplifying [tex]\(\sqrt{44}\)[/tex]:
[tex]\[
\sqrt{44} = \sqrt{4 \cdot 11} = \sqrt{4} \cdot \sqrt{11} = 2 \cdot \sqrt{11}
\][/tex]
5. Combining the results:
[tex]\[
\sqrt{-44} = i \cdot 2 \cdot \sqrt{11} = 2i \cdot \sqrt{11}
\][/tex]
6. Substituting back into the original expression:
[tex]\[
-5 - \sqrt{-44} = -5 - 2i \cdot \sqrt{11}
\][/tex]
Therefore, the value of [tex]\(-5 - \sqrt{-44}\)[/tex] simplifies to:
[tex]\[
-5 - 2i \cdot \sqrt{11}
\][/tex]
Comparing this to the provided options:
1. [tex]\( -5 - 2i \sqrt{11} \)[/tex]
2. [tex]\( -5 + 4 \sqrt{11i} \)[/tex]
3. [tex]\( -5 + 2 \sqrt{11i} \)[/tex]
4. [tex]\( -5 - 4i \sqrt{1.1} \)[/tex]
The correct option is:
[tex]\[
-5 - 2i \sqrt{11}
\][/tex]
