To find the square root of [tex]\(-100\)[/tex], we need to recognize that taking the square root of a negative number involves complex numbers. A complex number is typically written in the form [tex]\(a + bi\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are real numbers, and [tex]\(i\)[/tex] is the imaginary unit with the property that [tex]\(i^2 = -1\)[/tex].
Here are the detailed steps to find the square root of [tex]\(-100\)[/tex]:
1. Identify the negative number under the square root: We have [tex]\(\sqrt{-100}\)[/tex].
2. Rewrite the negative number in terms of the imaginary unit [tex]\(i\)[/tex]: We can express [tex]\(-100\)[/tex] as [tex]\(100 \cdot -1\)[/tex]. Remembering that [tex]\(-1 \cdot i^2 = -1\)[/tex], this can be rewritten as [tex]\(\sqrt{100 \cdot -1}\)[/tex].
3. Separate the terms under the square root: Using the property of square roots [tex]\(\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}\)[/tex], we get:
[tex]\[
\sqrt{100 \cdot -1} = \sqrt{100} \cdot \sqrt{-1}
\][/tex]
4. Calculate the square root of the positive number: The square root of 100 is 10. Hence:
[tex]\[
\sqrt{100} = 10
\][/tex]
5. Calculate the square root of [tex]\(-1\)[/tex]: By definition, the square root of [tex]\(-1\)[/tex] is [tex]\(i\)[/tex]. Therefore:
[tex]\[
\sqrt{-1} = i
\][/tex]
6. Combine the results: Multiply the results of the square roots obtained:
[tex]\[
\sqrt{100} \cdot \sqrt{-1} = 10 \cdot i = 10i
\][/tex]
7. Write the final answer in the form [tex]\(a + bi\)[/tex]: Since there is no real part in this case (real part is 0), we can write:
[tex]\[
0 + 10i
\][/tex]
Therefore, the square root of [tex]\(-100\)[/tex] is:
[tex]\[
\sqrt{-100} = 0 + 10i
\][/tex]
