Answer :
To solve the expression [tex]\(\sqrt{-98} + 71\)[/tex], we need to simplify [tex]\(\sqrt{-98}\)[/tex] involving imaginary numbers:
1. Identify the Imaginary Unit:
The expression [tex]\(\sqrt{-98}\)[/tex] involves a negative square root. We represent this as an imaginary number using [tex]\(i\)[/tex], where [tex]\(i = \sqrt{-1}\)[/tex]. Thus, [tex]\(\sqrt{-98}\)[/tex] is [tex]\(\sqrt{-1 \times 98}\)[/tex].
2. Separate the Negative Sign:
We write [tex]\(\sqrt{-98}\)[/tex] as [tex]\(\sqrt{-1} \times \sqrt{98}\)[/tex]. This becomes [tex]\(i \times \sqrt{98}\)[/tex].
3. Simplify [tex]\(\sqrt{98}\)[/tex]:
- Factor 98 to simplify its square root. Notice that 98 can be broken down into [tex]\(49 \times 2\)[/tex].
- Therefore, [tex]\(\sqrt{98} = \sqrt{49 \times 2} = \sqrt{49} \times \sqrt{2}\)[/tex].
- Since [tex]\(\sqrt{49} = 7\)[/tex], it simplifies further to [tex]\(7 \times \sqrt{2}\)[/tex].
4. Combine the Elements:
Substituting back, [tex]\(\sqrt{-98}\)[/tex] becomes [tex]\(i \times (7 \times \sqrt{2})\)[/tex], or simply [tex]\(7i \sqrt{2}\)[/tex].
5. Add to 71:
The original expression was [tex]\(\sqrt{-98} + 71\)[/tex]. Now we substitute to get [tex]\(71 + 7i \sqrt{2}\)[/tex].
Based on this simplification, the correct notation for [tex]\(\sqrt{-98} + 71\)[/tex] is:
[tex]\[71 + 7i \sqrt{2}\][/tex]
Therefore, the correct answer from the given options is [tex]\(71 + 7i \sqrt{2}\)[/tex].
1. Identify the Imaginary Unit:
The expression [tex]\(\sqrt{-98}\)[/tex] involves a negative square root. We represent this as an imaginary number using [tex]\(i\)[/tex], where [tex]\(i = \sqrt{-1}\)[/tex]. Thus, [tex]\(\sqrt{-98}\)[/tex] is [tex]\(\sqrt{-1 \times 98}\)[/tex].
2. Separate the Negative Sign:
We write [tex]\(\sqrt{-98}\)[/tex] as [tex]\(\sqrt{-1} \times \sqrt{98}\)[/tex]. This becomes [tex]\(i \times \sqrt{98}\)[/tex].
3. Simplify [tex]\(\sqrt{98}\)[/tex]:
- Factor 98 to simplify its square root. Notice that 98 can be broken down into [tex]\(49 \times 2\)[/tex].
- Therefore, [tex]\(\sqrt{98} = \sqrt{49 \times 2} = \sqrt{49} \times \sqrt{2}\)[/tex].
- Since [tex]\(\sqrt{49} = 7\)[/tex], it simplifies further to [tex]\(7 \times \sqrt{2}\)[/tex].
4. Combine the Elements:
Substituting back, [tex]\(\sqrt{-98}\)[/tex] becomes [tex]\(i \times (7 \times \sqrt{2})\)[/tex], or simply [tex]\(7i \sqrt{2}\)[/tex].
5. Add to 71:
The original expression was [tex]\(\sqrt{-98} + 71\)[/tex]. Now we substitute to get [tex]\(71 + 7i \sqrt{2}\)[/tex].
Based on this simplification, the correct notation for [tex]\(\sqrt{-98} + 71\)[/tex] is:
[tex]\[71 + 7i \sqrt{2}\][/tex]
Therefore, the correct answer from the given options is [tex]\(71 + 7i \sqrt{2}\)[/tex].
