To solve the expression [tex]\(\sqrt{-98}\)[/tex], we need to follow a series of steps that involve understanding the properties of square roots and imaginary numbers.
1. Identifying the Imaginary Unit:
We know that [tex]\(\sqrt{-1} = i\)[/tex], where [tex]\(i\)[/tex] is the imaginary unit.
2. Separating the Components:
Given the expression [tex]\(\sqrt{-98}\)[/tex], we can rewrite it as:
[tex]\[
\sqrt{-98} = \sqrt{98} \cdot \sqrt{-1} = \sqrt{98} \cdot i
\][/tex]
3. Simplifying the Square Root of 98:
Next, let's simplify [tex]\(\sqrt{98}\)[/tex]. We need to break this down into its prime factors:
[tex]\[
98 = 2 \times 49 = 2 \times 7^2
\][/tex]
Therefore,
[tex]\[
\sqrt{98} = \sqrt{2 \times 7^2}
\][/tex]
4. Using the Property of Square Roots:
We can further simplify using the property [tex]\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)[/tex]:
[tex]\[
\sqrt{98} = \sqrt{2 \times 7^2} = \sqrt{2} \times \sqrt{7^2} = \sqrt{2} \times 7
\][/tex]
5. Combining with the Imaginary Unit:
Substituting this back into our expression:
[tex]\[
\sqrt{-98} = \sqrt{98} \cdot i = 7 \sqrt{2} \cdot i = 7i\sqrt{2}
\][/tex]
Thus, the expression [tex]\(\sqrt{-98}\)[/tex] simplifies to [tex]\(7i\sqrt{2}\)[/tex].
Therefore, the correct option is:
[tex]\[
\boxed{7i\sqrt{2}}
\][/tex]
So the answer is C: [tex]\(7 i \sqrt{2}\)[/tex].
