To simplify [tex]\(\sqrt{-121}\)[/tex], let's proceed through the following steps:
1. Concept of Imaginary Numbers: The square root of a negative number involves the imaginary unit denoted by [tex]\(i\)[/tex], where [tex]\(i\)[/tex] is defined as [tex]\(\sqrt{-1}\)[/tex].
2. Separating the Negative Sign: We can start by expressing [tex]\(\sqrt{-121}\)[/tex] in terms of the imaginary unit:
[tex]\[
\sqrt{-121} = \sqrt{121 \times -1}
\][/tex]
3. Applying the Property of Square Roots: Using the property [tex]\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)[/tex], we can simplify further:
[tex]\[
\sqrt{121 \times -1} = \sqrt{121} \times \sqrt{-1}
\][/tex]
4. Simplifying Each Component:
- [tex]\(\sqrt{121}\)[/tex] is 11, because [tex]\(11 \times 11 = 121\)[/tex].
- [tex]\(\sqrt{-1}\)[/tex] is represented by [tex]\(i\)[/tex], the imaginary unit.
5. Combining the Results: Multiplying the results from the previous step:
[tex]\[
\sqrt{121} \times \sqrt{-1} = 11 \times i
\][/tex]
Thus, the simplified form of [tex]\(\sqrt{-121}\)[/tex] is:
[tex]\[
11i
\][/tex]
Out of the given options:
- [tex]\(-11i\)[/tex]
- [tex]\(11i\)[/tex]
- [tex]\(-11\)[/tex]
- [tex]\(11\)[/tex]
The correct answer is:
[tex]\[
\boxed{11i}
\][/tex]
