To determine which of the provided options is equivalent to [tex]\(\sqrt{-50}\)[/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 introduces the imaginary unit [tex]\(i\)[/tex], where [tex]\(i = \sqrt{-1}\)[/tex].
2. Factorizing the number inside the square root:
- Start with [tex]\(\sqrt{-50}\)[/tex].
- This can be written as [tex]\(\sqrt{-1 \times 50}\)[/tex].
3. Separation into imaginary and real components:
- [tex]\(\sqrt{-1 \times 50} = \sqrt{-1} \times \sqrt{50}\)[/tex].
- We know [tex]\(\sqrt{-1} = i\)[/tex], so this becomes [tex]\(i \times \sqrt{50}\)[/tex].
4. Simplifying the real part:
- [tex]\(\sqrt{50}\)[/tex] can be further simplified.
- [tex]\(50 = 25 \times 2\)[/tex], so [tex]\(\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5 \sqrt{2}\)[/tex].
5. Combining the imaginary and the simplified real part:
- Therefore, [tex]\(\sqrt{-50} = i \times 5 \sqrt{2} = 5i\sqrt{2}\)[/tex].
Given the work above, it is clear that [tex]\(\sqrt{-50}\)[/tex] is equivalent to [tex]\(5 i \sqrt{2}\)[/tex].
Thus, the correct answer is:
[tex]\[ 5 i \sqrt{2} \][/tex]
