To combine radicals using addition or subtraction, the radicals must have the same index and the same "radicand" (the expression underneath the radical). Here are the key points that must be true:
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Same Index: The radicals must be of the same type. For example, both must be square roots, cube roots, etc. If one radical is a square root and another is a cube root, they cannot be combined directly because their bases are fundamentally different.
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Same Radicand: The expressions under the radicals must be the same or must be simplified to the same form. For example, \(\sqrt{2}\) and \(3\sqrt{2}\) can be combined, but \(\sqrt{2}\) and \(\sqrt{3}\) cannot because 2 and 3 are different.
These conditions mean that, in many cases, it cannot be immediately determined if the radicals can be combined through addition or subtraction:
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Simplification: Sometimes, radicals might look different at first glance, but when simplified, they might actually be the same. For instance, \(\sqrt{8}\) can be simplified to \(2\sqrt{2}\), and \(\sqrt{2}\) would then be added or subtracted with \(2\sqrt{2}\) effectively if that sense happened in a larger expression.
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Expressions Involving Variables: When radicals involve variables, determining if the radicands are the same can be complicated due to the possibility of having different variable values. For instance, \(\sqrt{x}\) and \(\sqrt{4x}\) can generally not combine unless specific conditions on \(x\) are provided.
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Different Radicals: Even if they have the same index, different radicands (like \(\sqrt{5}\) and \(\sqrt{6}\)) cannot be combined, which might not be immediately obvious without a detailed inspection of the expressions.
In conclusion, careful examination of both the type and the expressions inside the radicals is necessary to determine if they can be combined through addition or subtraction.
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