The Difference Between Adding and Multiplying Like Radicals

📚 Understanding Like Radicals

Like radicals are radical expressions that have the same index and the same radicand. This means they can be combined through addition and subtraction, similar to how you combine like terms in algebra.

➕ Adding Like Radicals

Adding like radicals is similar to combining like terms. You simply add or subtract the coefficients (the numbers in front of the radical) while keeping the radical part the same.

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• 🔢 Definition: Addition of like radicals involves combining terms that have the same index and radicand.
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• 💡 Process: Identify like radicals, then add or subtract their coefficients. The radical part remains unchanged.
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• 📝 Example: $3\sqrt{2} + 5\sqrt{2} = (3+5)\sqrt{2} = 8\sqrt{2}$

✖️ Multiplying Like Radicals

Multiplying radicals involves multiplying the coefficients together and then multiplying the radicands together. If the indices are the same, you can combine the radicands under a single radical. Always simplify the resulting radical if possible.

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• 🔢 Definition: Multiplication of radicals involves multiplying the coefficients and radicands separately.
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• 💡 Process: Multiply coefficients, multiply radicands, and simplify the resulting radical.
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• 📝 Example: $3\sqrt{2} \times 5\sqrt{2} = (3 \times 5)\sqrt{2 \times 2} = 15\sqrt{4} = 15 \times 2 = 30$

📝 Comparison Table: Adding vs. Multiplying Like Radicals

🔑 Key Takeaways

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• 💡 Adding: Focus on combining coefficients of like radicals.
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• 📝 Multiplying: Multiply coefficients and radicands separately, then simplify.
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• 🧮 Simplification: Always simplify the radical after performing the operation.