Concept

Adding and Subtracting Like Square Roots

Like square roots are added and subtracted using the same procedure as combining like terms: add or subtract the coefficients while keeping the radical part unchanged. Just as 3x+8x=11x3x + 8x = 11x (the coefficients 33 and 88 are added while the variable xx stays the same), the expression 3x+8x=11x3\sqrt{x} + 8\sqrt{x} = 11\sqrt{x} is simplified by adding the coefficients 33 and 88 while the radical x\sqrt{x} remains unchanged. Similarly, 3+3=23\sqrt{3} + \sqrt{3} = 2\sqrt{3}, because a single 3\sqrt{3} has an implicit coefficient of 11, and 1+1=21 + 1 = 2.

Square roots with different radicands cannot be combined. The expression 2+7\sqrt{2} + \sqrt{7} must be left as is — it is a common error to incorrectly merge the two radicals into 2+7\sqrt{2 + 7}. In fact, 2+79\sqrt{2} + \sqrt{7} \neq \sqrt{9}, because the radical sign acts as a grouping symbol and only governs the expression directly underneath it.

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Updated 2026-04-21

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