To express [tex]\(\sqrt{-50}\)[/tex] in terms of [tex]\(i\)[/tex], let's go through the steps one by one.
1. Identify the negative number under the square root:
[tex]\[
\sqrt{-50}
\][/tex]
2. Recognize that the square root of a negative number can be expressed in terms of the imaginary unit [tex]\(i\)[/tex]:
By definition, [tex]\(i\)[/tex] is the imaginary unit where [tex]\(i^2 = -1\)[/tex]. Hence, [tex]\(\sqrt{-1} = i\)[/tex].
3. Separate the square root of the negative number:
We can break this down into the product of two square roots:
[tex]\[
\sqrt{-50} = \sqrt{-1 \times 50} = \sqrt{-1} \times \sqrt{50}
\][/tex]
4. Simplify using the imaginary unit [tex]\(i\)[/tex]:
Substitute [tex]\(\sqrt{-1}\)[/tex] with [tex]\(i\)[/tex]:
[tex]\[
\sqrt{-50} = i \times \sqrt{50}
\][/tex]
5. Calculate the square root of the positive number:
[tex]\[
\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}
\][/tex]
6. Combine the results:
[tex]\[
\sqrt{-50} = i \times 5\sqrt{2} = 5\sqrt{2}i
\][/tex]
7. Evaluate the numerical value:
We know that [tex]\(\sqrt{50} \approx 7.0710678118654755\)[/tex]. Hence:
[tex]\[
\sqrt{-50} = 7.0710678118654755i
\][/tex]
8. Final simplified answer:
[tex]\[
\boxed{7.0710678118654755i}
\][/tex]
So, the expression [tex]\(\sqrt{-50}\)[/tex] in terms of [tex]\(i\)[/tex] is [tex]\(7.0710678118654755i\)[/tex].
