Answer :
To find an equivalent expression to [tex]\(\sqrt{-20}\)[/tex], we need to deal with the square root of a negative number, which involves the imaginary unit, [tex]\(i\)[/tex].
1. Understand the Imaginary Unit:
- The imaginary unit [tex]\(i\)[/tex] is defined as [tex]\(\sqrt{-1}\)[/tex].
- Therefore, any square root of a negative number can be expressed in terms of [tex]\(i\)[/tex].
2. Break Down the Expression:
- The expression [tex]\(\sqrt{-20}\)[/tex] can be broken down into:
[tex]\[
\sqrt{-20} = \sqrt{-1} \times \sqrt{20}
\][/tex]
3. Express with the Imaginary Unit:
- We know that [tex]\(\sqrt{-1} = i\)[/tex].
- So, [tex]\(\sqrt{-20}\)[/tex] can be rewritten as:
[tex]\[
\sqrt{-20} = i \times \sqrt{20}
\][/tex]
4. Calculate [tex]\(\sqrt{20}\)[/tex]:
- [tex]\(\sqrt{20}\)[/tex] is the positive square root of 20.
- [tex]\(\sqrt{20}\)[/tex] is approximately 4.4721 (rounded to four decimal places).
5. Combine the Results:
- Multiply the square root by the imaginary unit:
[tex]\[
\sqrt{-20} = 4.4721i
\][/tex]
The expression [tex]\(\sqrt{-20}\)[/tex] is equivalent to approximately [tex]\(4.4721i\)[/tex]. The given option [tex]\(20i\)[/tex] is not equivalent to [tex]\(\sqrt{-20}\)[/tex]. Thus, the correct equivalent expression for [tex]\(\sqrt{-20}\)[/tex] is [tex]\(4.4721i\)[/tex], not [tex]\(20i\)[/tex].
1. Understand the Imaginary Unit:
- The imaginary unit [tex]\(i\)[/tex] is defined as [tex]\(\sqrt{-1}\)[/tex].
- Therefore, any square root of a negative number can be expressed in terms of [tex]\(i\)[/tex].
2. Break Down the Expression:
- The expression [tex]\(\sqrt{-20}\)[/tex] can be broken down into:
[tex]\[
\sqrt{-20} = \sqrt{-1} \times \sqrt{20}
\][/tex]
3. Express with the Imaginary Unit:
- We know that [tex]\(\sqrt{-1} = i\)[/tex].
- So, [tex]\(\sqrt{-20}\)[/tex] can be rewritten as:
[tex]\[
\sqrt{-20} = i \times \sqrt{20}
\][/tex]
4. Calculate [tex]\(\sqrt{20}\)[/tex]:
- [tex]\(\sqrt{20}\)[/tex] is the positive square root of 20.
- [tex]\(\sqrt{20}\)[/tex] is approximately 4.4721 (rounded to four decimal places).
5. Combine the Results:
- Multiply the square root by the imaginary unit:
[tex]\[
\sqrt{-20} = 4.4721i
\][/tex]
The expression [tex]\(\sqrt{-20}\)[/tex] is equivalent to approximately [tex]\(4.4721i\)[/tex]. The given option [tex]\(20i\)[/tex] is not equivalent to [tex]\(\sqrt{-20}\)[/tex]. Thus, the correct equivalent expression for [tex]\(\sqrt{-20}\)[/tex] is [tex]\(4.4721i\)[/tex], not [tex]\(20i\)[/tex].