Question
Write in terms of \( i \). Simplify your answer as much as possible. \( \sqrt{-48} \)
Ask by Dunn Tucker. 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 \( i \), follow these steps:
1. **Separate the negative sign:**
\[
\sqrt{-48} = \sqrt{-1 \times 48} = \sqrt{-1} \times \sqrt{48}
\]
2. **Express the square root of -1 as \( i \):**
\[
\sqrt{-1} = i
\]
So,
\[
\sqrt{-48} = i \times \sqrt{48}
\]
3. **Simplify \( \sqrt{48} \):**
\[
48 = 16 \times 3 \quad \Rightarrow \quad \sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3}
\]
4. **Combine the results:**
\[
\sqrt{-48} = i \times 4\sqrt{3} = 4\sqrt{3} \, i
\]
**Final Answer:**
\[
4\,\sqrt{3}\,i
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To simplify \( \sqrt{-48} \), we start by extracting the negative sign: \[ \sqrt{-48} = \sqrt{48} \cdot \sqrt{-1} = \sqrt{48} \cdot i \] Next, we simplify \( \sqrt{48} \). We can express 48 as \( 16 \times 3 \): \[ \sqrt{48} = \sqrt{16 \cdot 3} = \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3} \] Putting it all together, we have: \[ \sqrt{-48} = 4\sqrt{3} \cdot i \] Thus, the simplified form of \( \sqrt{-48} \) is: \[ 4\sqrt{3}i \]
