To simplify the expression \(-2i \sqrt{-12}\), follow these steps:
1. Break Down the Square Root of the Negative Number:
Recognize that \( \sqrt{-12} \) can be broken down as \( \sqrt{12 \cdot -1} \). This can be further simplified using the property of square roots:
[tex]\[
\sqrt{12 \cdot -1} = \sqrt{12} \cdot \sqrt{-1}
\][/tex]
2. Simplify the Imaginary Unit:
Recall that \( \sqrt{-1} \) is defined as \( i \). So,
[tex]\[
\sqrt{-12} = \sqrt{12} \cdot i
\][/tex]
3. Substitute Back into the Expression:
Substitute \( \sqrt{-12} = \sqrt{12} \cdot i \) back into the original expression:
[tex]\[
-2i \sqrt{-12} = -2i (\sqrt{12} \cdot i)
\][/tex]
4. Combine Like Terms:
When you multiply \( -2i \) and \( \sqrt{12} \cdot i \), you focus first on multiplying the imaginary units:
[tex]\[
-2i \cdot i \cdot \sqrt{12}
\][/tex]
Given that \( i \cdot i = i^2 \), and knowing that \( i^2 = -1 \):
[tex]\[
-2i^2 \cdot \sqrt{12} = -2 \cdot (-1) \cdot \sqrt{12} = 2 \cdot \sqrt{12}
\][/tex]
5. Simplify the Radicand:
The term \( \sqrt{12} \) can be further simplified by recognizing that \( 12 = 4 \cdot 3 \), and \( 4 \) is a perfect square:
[tex]\[
\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}
\][/tex]
6. Combine All Parts:
Substitute \( \sqrt{12} = 2\sqrt{3} \) back into the expression \( 2 \cdot \sqrt{12} \):
[tex]\[
2 \cdot \sqrt{12} = 2 \cdot 2\sqrt{3} = 4\sqrt{3}
\][/tex]
So, the simplified form of the expression \(-2i \sqrt{-12}\) is:
[tex]\[
4\sqrt{3}
\][/tex]
