What conditions must be met to combine radicals through addition or subtraction?

Answer 1

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.