Question
Write in terms of \( i \). Simplify your answer as much as possible. \[ \sqrt{-80} \]
Ask by Medina Carroll. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
\sqrt{-80} = 4\sqrt{5}\,i
\]
Solution
To simplify \(\sqrt{-80}\) in terms of \(i\), follow these steps:
1. **Express the square root of a negative number using \(i\):**
\[
\sqrt{-80} = \sqrt{-1 \times 80} = \sqrt{-1} \times \sqrt{80} = i \times \sqrt{80}
\]
2. **Simplify \(\sqrt{80}\):**
\[
\sqrt{80} = \sqrt{16 \times 5} = \sqrt{16} \times \sqrt{5} = 4\sqrt{5}
\]
3. **Combine the results:**
\[
\sqrt{-80} = i \times 4\sqrt{5} = 4\sqrt{5}\,i
\]
**Final Answer:**
\[
\sqrt{-80} = 4\,\sqrt{5}\,i
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To express \( \sqrt{-80} \) in terms of \( i \), we first rewrite the expression as follows: \[ \sqrt{-80} = \sqrt{80} \cdot \sqrt{-1} = \sqrt{80} \cdot i \] Next, we simplify \( \sqrt{80} \). We can break it down: \[ \sqrt{80} = \sqrt{16 \cdot 5} = \sqrt{16} \cdot \sqrt{5} = 4\sqrt{5} \] Now, substituting back into our expression gives us: \[ \sqrt{-80} = 4\sqrt{5} \cdot i \] Therefore, the simplified form of \( \sqrt{-80} \) in terms of \( i \) is: \[ \sqrt{-80} = 4\sqrt{5}i \]
