Here are two key differences between the procedures for multiplying radicals and the procedures for adding and subtracting them:
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Combining versus Simplifying:
- Addition/Subtraction: When adding or subtracting radicals, you can only combine them if they have the same radicand (the number inside the radical). For example, \( \sqrt{2} + 3\sqrt{2} = 4\sqrt{2} \) because they have the same radicand. However, \( \sqrt{2} + \sqrt{3} \) cannot be combined further and remains \( \sqrt{2} + \sqrt{3} \).
- Multiplication: When multiplying radicals, you can simply multiply the radicands regardless of whether they are the same or not. For example, \( \sqrt{2} \times \sqrt{3} = \sqrt{6} \). The multiplication of radicals follows the property \( \sqrt{a} \times \sqrt{b} = \sqrt{a \cdot b} \).
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Distribution versus Factorization:
- Addition/Subtraction: In addition or subtraction, the radicals are treated as terms that can be rearranged, but their individual components remain separated. Combining them often requires factoring out common terms, like factoring a \( \sqrt{2} \) out from \( \sqrt{2} + 3\sqrt{2} \).
- Multiplication: In multiplication, you apply the distributive property, which often involves changing the form of the radicals. For example, \( (\sqrt{2} + \sqrt{3})(\sqrt{2} - \sqrt{3}) \) results in a simplified form via the difference of squares: \( 2 - 3 = -1 \).
These key differences highlight the distinct approaches needed for handling operations with radicals.
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