To solve for [tex]\(\sqrt{-121}\)[/tex], let’s break it down step by step.
1. Determine the Nature of the Number Inside the Square Root:
- The number inside the square root is [tex]\(-121\)[/tex], which is negative. The square root of a negative number involves imaginary numbers.
2. Understand the Concept of Imaginary Numbers:
- By definition, [tex]\(i\)[/tex] (the imaginary unit) satisfies [tex]\(i^2 = -1\)[/tex].
- Therefore, [tex]\(\sqrt{-1} = i\)[/tex].
3. Simplify the Square Root of the Negative Number:
- We can rewrite [tex]\(\sqrt{-121}\)[/tex] as [tex]\(\sqrt{121 \cdot (-1)}\)[/tex].
- This can be further broken down into [tex]\(\sqrt{121} \cdot \sqrt{-1}\)[/tex].
4. Calculate the Square Roots Separately:
- [tex]\(\sqrt{121}\)[/tex] is straightforward: [tex]\(\sqrt{121} = 11\)[/tex] because [tex]\(11^2 = 121\)[/tex].
- [tex]\(\sqrt{-1} = i\)[/tex], by the definition of the imaginary unit.
5. Combine the Results:
- Combining these, we get [tex]\(\sqrt{-121} = 11 \cdot i\)[/tex].
Thus, [tex]\(\sqrt{-121} = 11i\)[/tex].
The equivalent choice is:
A. [tex]\(11i\)[/tex]
