Question
Use the imaginary number \( i \) to rewrite the expression below as a complex number. Simplify all radicals. \[ \sqrt{-12} \]
Ask by Ball Knight. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The expression \(\sqrt{-12}\) as a complex number is \(2\sqrt{3}i\).
Solution
To rewrite the expression \(\sqrt{-12}\) as a complex number using the imaginary unit \(i\), follow these steps:
1. **Separate the negative sign:**
\[
\sqrt{-12} = \sqrt{12} \times \sqrt{-1}
\]
2. **Express \(\sqrt{-1}\) as \(i\):**
\[
\sqrt{-12} = \sqrt{12} \times i
\]
3. **Simplify \(\sqrt{12}\):**
\[
\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}
\]
4. **Combine the results:**
\[
\sqrt{-12} = 2\sqrt{3} \, i
\]
So, the expression \(\sqrt{-12}\) rewritten as a complex number is:
\[
2\,\sqrt{3}\,i
\]
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Simplify this solution Extra Insights
To rewrite \( \sqrt{-12} \) as a complex number, we first separate the negative part from the positive part. Recall that \( \sqrt{-1} = i \). We can express \( \sqrt{-12} \) as: \[ \sqrt{-12} = \sqrt{12} \cdot \sqrt{-1} = \sqrt{12} \cdot i \] Next, we simplify \( \sqrt{12} \): \[ \sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \] Now, substituting back, we get: \[ \sqrt{-12} = 2\sqrt{3} \cdot i \] Thus, the expression simplified as a complex number is: \[ \sqrt{-12} = 2\sqrt{3}i \]
