At Westonci.ca, we provide clear, reliable answers to all your questions. Join our vibrant community and get the solutions you need. Connect with a community of experts ready to provide precise solutions to your questions quickly and accurately. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
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]
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]
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.