Question
What conditions must be met to combine radicals through addition or subtraction?
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GPT-4o mini
To combine radicals through addition or subtraction, the following conditions must be met:
1. **Like Radicals**: The radicals must be like radicals, meaning they have the same index and the same radicand (the number or expression inside the radical). For example, \( \sqrt{3} \) and \( 2\sqrt{3} \) are like radicals because they both involve \( \sqrt{3} \). However, \( \sqrt{3} \) and \( \sqrt{5} \) cannot be combined because they have different radicands.
2. **Coefficients**: Even if the radicals are like, you can only combine them by adding or subtracting their coefficients. For instance, you can combine \( 3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2} \) because they are like radicals. On the other hand, \( 3\sqrt{2} - 2\sqrt{2} = 1\sqrt{2} \) simplifies to \( \sqrt{2} \).
3. **Simplification**: Sometimes, radicals can be simplified before combining. For instance, \( \sqrt{8} \) can be simplified to \( 2\sqrt{2} \), allowing it to be combined with \( 3\sqrt{2} \).
To summarize, ensure that the radicals you want to combine are like radicals (same index and radicand) and then add or subtract their coefficients accordingly.
1. **Like Radicals**: The radicals must be like radicals, meaning they have the same index and the same radicand (the number or expression inside the radical). For example, \( \sqrt{3} \) and \( 2\sqrt{3} \) are like radicals because they both involve \( \sqrt{3} \). However, \( \sqrt{3} \) and \( \sqrt{5} \) cannot be combined because they have different radicands.
2. **Coefficients**: Even if the radicals are like, you can only combine them by adding or subtracting their coefficients. For instance, you can combine \( 3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2} \) because they are like radicals. On the other hand, \( 3\sqrt{2} - 2\sqrt{2} = 1\sqrt{2} \) simplifies to \( \sqrt{2} \).
3. **Simplification**: Sometimes, radicals can be simplified before combining. For instance, \( \sqrt{8} \) can be simplified to \( 2\sqrt{2} \), allowing it to be combined with \( 3\sqrt{2} \).
To summarize, ensure that the radicals you want to combine are like radicals (same index and radicand) and then add or subtract their coefficients accordingly.
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