To determine the equivalent expression to [tex]\(\sqrt{-18}\)[/tex], let's break it down step-by-step.
1. Understanding the Square Root of a Negative Number:
The square root of a negative number involves the imaginary unit [tex]\(i\)[/tex], where [tex]\(i = \sqrt{-1}\)[/tex]. Hence,
[tex]\[
\sqrt{-a} = i \sqrt{a}
\][/tex]
2. Applying this to Our Expression:
Given the expression [tex]\(\sqrt{-18}\)[/tex], we can rewrite it as:
[tex]\[
\sqrt{-18} = i \sqrt{18}
\][/tex]
3. Simplifying [tex]\(\sqrt{18}\)[/tex]:
To simplify [tex]\(\sqrt{18}\)[/tex], we factorize 18 into its prime factors:
[tex]\[
18 = 9 \times 2
\][/tex]
Thus,
[tex]\[
\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3 \sqrt{2}
\][/tex]
4. Combining the Results:
Substituting back, we get:
[tex]\[
\sqrt{-18} = i \sqrt{18} = i \cdot (3 \sqrt{2}) = 3i \sqrt{2}
\][/tex]
5. Conclusion:
Therefore, the expression [tex]\(\sqrt{-18}\)[/tex] is equivalent to [tex]\(3i \sqrt{2}\)[/tex].
Given this breakdown, the correct choice is:
B. [tex]\(3 i \sqrt{2}\)[/tex]
