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Example

Simplifying 9-\sqrt{9} and 144-\sqrt{144}

To simplify an expression of the form n-\sqrt{n} where nn is a perfect square, first evaluate the principal (positive) square root, then apply the negative sign in front:

9=3-\sqrt{9} = -3: The negative sign sits in front of the radical, not underneath it. Evaluate the square root first: 9=3\sqrt{9} = 3 because 32=93^2 = 9. Then apply the negation: 3-3.

144=12-\sqrt{144} = -12: Again, the negative sign is in front of the radical. Evaluate the square root first: 144=12\sqrt{144} = 12 because 122=14412^2 = 144. Then apply the negation: 12-12.

In both cases the procedure is the same: recognize the radicand as a perfect square, find its principal square root, and then take the opposite of that result. The key observation is that the negative sign is not part of the radicand — it is applied after the square root has been computed.

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Updated 2026-04-21

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