To find the value of [tex]\(\sqrt{-100}\)[/tex], we need to recognize that we are dealing with the square root of a negative number. The square root of a negative number involves imaginary numbers. Specifically, we use the imaginary unit [tex]\(i\)[/tex], where [tex]\(i = \sqrt{-1}\)[/tex].
Let's go through the steps:
1. Start with the given expression:
[tex]\[
\sqrt{-100}
\][/tex]
2. Rewrite the negative number using the imaginary unit [tex]\(i\)[/tex]:
[tex]\[
\sqrt{-100} = \sqrt{100 \cdot (-1)} = \sqrt{100} \cdot \sqrt{-1}
\][/tex]
3. Simplify the square root of 100:
[tex]\[
\sqrt{100} = 10
\][/tex]
4. Combine the results with the imaginary unit [tex]\(i\)[/tex]:
[tex]\[
\sqrt{-1} = i
\][/tex]
Therefore:
[tex]\[
\sqrt{100} \cdot \sqrt{-1} = 10 \cdot i = 10i
\][/tex]
Hence, the value of [tex]\(\sqrt{-100}\)[/tex] is:
[tex]\[
\sqrt{-100} = 10i
\][/tex]
Additionally, if we want to express this in terms of its real and imaginary parts, we have:
- The real part: [tex]\(0\)[/tex]
- The imaginary part: [tex]\(10\)[/tex]
Thus, the complete expression of [tex]\(\sqrt{-100}\)[/tex] is:
[tex]\[
\sqrt{-100} = 0 + 10i
\][/tex]
Or simply:
[tex]\[
\sqrt{-100} = 10i
\][/tex]
