You're correct! In order to add or subtract radicals, the radicands (the numbers under the square root symbol) must be the same, resulting in what we call "like radicals." For instance, you can combine \( \sqrt{2} \) and \( \sqrt{2} \) to get \( 2\sqrt{2} \), but you cannot combine \( \sqrt{2} \) and \( \sqrt{3} \) because they have different radicands.
Here's a brief breakdown:
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Like Radicals: These have the same radicand. For example: \[ 3\sqrt{2} + 2\sqrt{2} = (3 + 2)\sqrt{2} = 5\sqrt{2} \]
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Unlike Radicals: These have different radicands and cannot be directly combined. For example: \[ \sqrt{2} + \sqrt{3} \text{ cannot be simplified further.} \]
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Rational and Irrational Numbers: While the square root of some numbers (like perfect squares) can be expressed as rational numbers (e.g., \( \sqrt{4} = 2 \)), many square roots are irrational and cannot be precisely expressed as fractions or finite decimals (e.g., \( \sqrt{2} \approx 1.414213... \)).
In summary, combining radicals through addition or subtraction is only feasible when they are like radicals, i.e., they share the same radicand.