Two key differences between the procedures for multiplying radicals and the procedures for adding or subtracting them are:
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Combining Like Terms:
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Adding or Subtracting Radicals: To add or subtract radicals, the terms must be like terms, meaning they must have the same radicand (the number under the radical), as well as the same index (for example, both square roots or both cube roots). Only the coefficients (the numbers in front) can be combined. For example, \(3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}\) because \( \sqrt{2} \) is the same in both terms.
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Multiplying Radicals: When multiplying radicals, you can directly multiply the coefficients and the radicands, regardless of whether they are like terms. For example, \(3\sqrt{2} \cdot 5\sqrt{3} = 15\sqrt{6}\) because you multiply \(3\) and \(5\) to get \(15\) and \(\sqrt{2} \cdot \sqrt{3} = \sqrt{6}\).
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Use of Distribution:
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Adding or Subtracting Radicals: When adding or subtracting radicals, you do not distribute the radical across terms. Instead, you simply combine like terms if possible and retain the individual terms that cannot be combined. For example, \( \sqrt{5} + \sqrt{7} \) remains as is because the terms are not like.
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Multiplying Radicals: When multiplying, you often use the distributive property (also known as the FOIL method when dealing with binomials). For example, when you multiply \( (2 + \sqrt{3})(3 + \sqrt{2}) \), you would use distribution to find each product: \( 2 \cdot 3 + 2\sqrt{2} + 3\sqrt{3} + \sqrt{6} \).
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These differences highlight how the properties and conditions of radicals affect the operations of addition, subtraction, and multiplication.
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