To simplify the expression [tex]\(-5 - \sqrt{-44}\)[/tex], follow these steps:
1. Understand the nature of the square root: The term [tex]\(\sqrt{-44}\)[/tex] involves the square root of a negative number, which results in an imaginary number. Recall that [tex]\(\sqrt{-1} = i\)[/tex], where [tex]\(i\)[/tex] is the imaginary unit.
2. Break down the expression under the square root: We have:
[tex]\[
\sqrt{-44} = \sqrt{-1 \cdot 44}
\][/tex]
3. Use the property of square roots: The property [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex] allows us to rewrite the expression as:
[tex]\[
\sqrt{-44} = \sqrt{-1} \cdot \sqrt{44}
\][/tex]
4. Substitute the imaginary unit [tex]\(i\)[/tex]: Thus:
[tex]\[
\sqrt{-44} = i \cdot \sqrt{44}
\][/tex]
5. Simplify [tex]\(\sqrt{44}\)[/tex]: Notice that 44 can be decomposed into factors to further simplify the expression:
[tex]\[
\sqrt{44} = \sqrt{4 \cdot 11} = \sqrt{4} \cdot \sqrt{11} = 2 \cdot \sqrt{11}
\][/tex]
6. Combine the simplified terms: Therefore:
[tex]\[
\sqrt{-44} = i \cdot 2 \sqrt{11} = 2i \sqrt{11}
\][/tex]
7. Substitute back into the original expression:
[tex]\[
-5 - \sqrt{-44} = -5 - 2i \sqrt{11}
\][/tex]
Thus, the fully simplified form of the expression is:
[tex]\[
-5 - 2i \sqrt{11}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{-5 - 2i \sqrt{11}}
\][/tex]
