1Cademy - Simplifying $$\sqrt{25} + \sqrt{144}$$ versus $$\sqrt{25 + 144}$$

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Radical Sign as a Grouping Symbol

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Simplifying $\sqrt{25} + \sqrt{144}$ versus $\sqrt{25 + 144}$

This example simplifies two expressions that use the same numbers but differ in whether the addition is inside or outside the radical, demonstrating that the results are not equal. ⓐ $\sqrt{25} + \sqrt{144} = 17$ : Each radical contains only a single number, so evaluate each square root separately using the order of operations: $\sqrt{25} = 5$ and $\sqrt{144} = 12$ . Then add: $5 + 12 = 17$ . ⓑ $\sqrt{25 + 144} = 13$ : Because the radical sign acts as a grouping symbol, simplify the expression under the radical first: $25 + 144 = 169$ . Then take the square root: $\sqrt{169} = 13$ . The two results — $17$ and $13$ — are different, which demonstrates that $\sqrt{a} + \sqrt{b}$ is generally not equal to $\sqrt{a + b}$ . The placement of the addition — outside or inside the radical — fundamentally changes which operation is performed first and therefore changes the outcome.

Updated 2026-06-25

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