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Use the product rule to simplify the radical. \[ 3 \sqrt{12} \]

Ask by Weaver Carlson. in the United States
Jan 22,2025

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\( 6 \sqrt{3} \)

Solución

To simplify \( 3 \sqrt{12} \) using the product rule for radicals, follow these steps: 1. **Factor the Radicand:** \[ 12 = 4 \times 3 \] 2. **Apply the Product Rule for Radicals:** \[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} \] 3. **Simplify the Square Root:** \[ \sqrt{4} = 2 \] So, \[ \sqrt{12} = 2 \sqrt{3} \] 4. **Multiply by the Coefficient:** \[ 3 \sqrt{12} = 3 \times 2 \sqrt{3} = 6 \sqrt{3} \] **Final Simplified Form:** \[ 6 \sqrt{3} \]

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Mind Expander

First, let’s simplify \( \sqrt{12} \). We can break it down as follows: \( \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \). Therefore, we can substitute this back into the original expression: \[ 3 \sqrt{12} = 3 \times 2\sqrt{3} = 6\sqrt{3}. \] So, \( 3 \sqrt{12} \) simplifies to \( 6\sqrt{3} \). Now we've got our final result! It’s like unlocking an algebraic treasure with just a few steps!

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