Question
Write in terms of \( i \). Simplify your answer as much as possible. \[ -\sqrt{-48} \]
Ask by Lang Bates. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\(-4\sqrt{3}\,i\)
Solution
To simplify \(-\sqrt{-48}\) in terms of \(i\) (where \(i = \sqrt{-1}\)), follow these steps:
1. **Express the square root of a negative number using \(i\):**
\[
\sqrt{-48} = \sqrt{48} \cdot \sqrt{-1} = \sqrt{48} \cdot i
\]
2. **Simplify \(\sqrt{48}\):**
\[
\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3}
\]
3. **Combine the results:**
\[
\sqrt{-48} = 4\sqrt{3} \cdot i
\]
4. **Include the negative sign:**
\[
-\sqrt{-48} = -4\sqrt{3} \cdot i
\]
**Final Simplified Form:**
\[
-4\sqrt{3}\,i
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To simplify \(-\sqrt{-48}\), we start by separating the negative sign from the square root: \[ -\sqrt{-48} = -\sqrt{48} \cdot \sqrt{-1} \] We know that \(\sqrt{-1} = i\), so we rewrite the expression as: \[ -\sqrt{48} \cdot i \] Next, we simplify \(\sqrt{48}\). The number 48 can be factored into \(16 \cdot 3\), where 16 is a perfect square: \[ \sqrt{48} = \sqrt{16 \cdot 3} = \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3} \] Putting this back into the expression, we have: \[ -\sqrt{-48} = -4\sqrt{3} \cdot i \] Thus, the simplified answer is: \[ -4\sqrt{3}i \]
