To simplify the expression [tex]\(\sqrt{-50}\)[/tex], follow these steps:
1. Recognize the format: Understand that we are dealing with the square root of a negative number, which will involve imaginary units.
2. Decompose the expression: Rewrite the square root of [tex]\(-50\)[/tex] using the property of square roots for negative numbers:
[tex]\[
\sqrt{-50} = \sqrt{-1 \times 50}
\][/tex]
3. Introduce the imaginary unit: Recall that [tex]\(\sqrt{-1} = i\)[/tex]. Using this, we can separate the imaginary part from the positive part:
[tex]\[
\sqrt{-50} = \sqrt{-1} \times \sqrt{50} = i \times \sqrt{50}
\][/tex]
4. Simplify the square root: Break down [tex]\(\sqrt{50}\)[/tex] further. Notice that [tex]\(50\)[/tex] can be factored into [tex]\(25 \times 2\)[/tex]:
[tex]\[
\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2}
\][/tex]
5. Evaluate the components: Since [tex]\(\sqrt{25} = 5\)[/tex], substitute this value back into the expression:
[tex]\[
\sqrt{50} = 5 \times \sqrt{2}
\][/tex]
6. Combine the results: Now that we have [tex]\(\sqrt{50}\)[/tex] simplified, substitute it back into the original equation:
[tex]\[
\sqrt{-50} = i \times 5 \times \sqrt{2} = 5i \sqrt{2}
\][/tex]
So, the simplified form of [tex]\(\sqrt{-50}\)[/tex] is [tex]\(5i \sqrt{2}\)[/tex].
Therefore, the correct answer is:
[tex]\[ 5 i \sqrt{2} \][/tex]
