1Cademy - What makes two square roots like, and how do you add or subtract them?

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Adding and Subtracting Like Square Roots

Essay

What makes two square roots "like," and how do you add or subtract them?

Question: What makes two square roots "like," and how do you add or subtract them?

Sample answer: Two square roots are "like" when they have the same radicand (the number under the radical sign). To add or subtract like square roots, add or subtract the coefficients and keep the radicand unchanged: $3\sqrt{x} + 8\sqrt{x} = 11\sqrt{x}$ . A square root with no visible coefficient has an implicit coefficient of $1$ , so $\sqrt{3} + \sqrt{3} = 2\sqrt{3}$ . If the radicands are different, the square roots cannot be combined: $\sqrt{2} + \sqrt{7}$ must stay as is, since $\sqrt{2} + \sqrt{7} \neq \sqrt{9}$ .

Key points:

Like square roots have the same radicand.
Only the coefficients are added or subtracted.
The radicand does not change.
A square root with no coefficient has an implicit coefficient of 1.
Square roots with different radicands cannot be combined.

Rubric: Full credit: states that like square roots share the same radicand, explains that only coefficients are added or subtracted while the radicand stays the same, and notes that different radicands cannot be combined. Partial credit: covers some but not all of these points. No credit: incorrect or missing explanation.

Updated 2026-07-10

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