Here are two key differences between how radicals are multiplied and how they are added or subtracted:
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Combination of Terms:
- Multiplication of Radicals: When multiplying radicals, you can directly multiply the numbers under the radical signs. For example, \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\). This means you can combine the radicands (the numbers inside the radicals) when multiplying.
- Addition/Subtraction of Radicals: When adding or subtracting radicals, you can only combine them if they have the same radicand (i.e., they are like terms). For example, \(\sqrt{a} + \sqrt{a} = 2\sqrt{a}\), but \(\sqrt{a} + \sqrt{b}\) cannot be simplified into a single radical unless \(a = b\).
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Distribution:
- Multiplication of Radicals: The distributive property applies, which allows for the multiplication of each term separately. For example, \(\sqrt{a} (\sqrt{b} + \sqrt{c}) = \sqrt{a}\sqrt{b} + \sqrt{a}\sqrt{c}\).
- Addition/Subtraction of Radicals: The distributive property does not apply in the same way when adding or subtracting. For instance, \(\sqrt{a} + (\sqrt{b} + \sqrt{c})\) cannot be simplified by distributing \(\sqrt{a}\) to all terms. The radicals in an addition or subtraction operation must be kept separate unless they can be combined according to the rules of like terms.