To determine the equivalent expression for [tex]\(\sqrt{-121}\)[/tex], we need to find the square root of a negative number.
First, recall that the square root of a negative number involves the imaginary unit [tex]\(i\)[/tex], where [tex]\(i\)[/tex] is defined as [tex]\(\sqrt{-1}\)[/tex].
Given the expression:
[tex]\[
\sqrt{-121}
\][/tex]
We can break this down using the property of square roots for negative numbers:
[tex]\[
\sqrt{-121} = \sqrt{121 \cdot (-1)}
\][/tex]
Using the property of square roots that [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex], we get:
[tex]\[
\sqrt{121 \cdot (-1)} = \sqrt{121} \cdot \sqrt{-1}
\][/tex]
We know:
[tex]\[
\sqrt{121} = 11 \quad \text{and} \quad \sqrt{-1} = i
\][/tex]
Therefore:
[tex]\[
\sqrt{-121} = 11 \cdot i = 11i
\][/tex]
Now, we need to determine which of the choices matches this result. The expression [tex]\(11i\)[/tex] corresponds to the imaginary part of the number.
Reviewing the choices:
A. [tex]\(-\sqrt{11}\)[/tex] is a real number and does not involve [tex]\(i\)[/tex].
B. [tex]\(-\sqrt{11 i}\)[/tex] is not equivalent to our result since it involves a square root of an imaginary unit.
C. [tex]\(\sqrt{11 i}\)[/tex] is not equivalent to our result and improperly combines [tex]\(i\)[/tex] inside the square root.
D. [tex]\(11/\)[/tex] is not mathematically valid notation and incomplete.
E. [tex]\(-11\)[/tex] is a real number and does not involve [tex]\(i\)[/tex].
None of the choices directly correspond to [tex]\(11i\)[/tex].
Thus, the best choice equivalent to [tex]\(11i\)[/tex] from the given options would technically be none, as none match exactly. However, we established that the correct simplified and mathematical equivalent of [tex]\(\sqrt{-121}\)[/tex] is:
[tex]\[
11i
\][/tex]
