1Cademy - Identifying $$\sqrt{-169}$$ and $$-\sqrt{64}$$ as Real or Not Real

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Square Root of a Negative Number Is Not Real

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Identifying $\sqrt{-169}$ and $-\sqrt{64}$ as Real or Not Real

Determine whether each expression represents a real number.

Part (a) — $\sqrt{-169}$ : Ask whether any real number squared equals $-169$ . Since every real number squared is non-negative, no real number satisfies this condition. Therefore $\sqrt{-169}$ is not a real number.

Part (b) — $-\sqrt{64}$ : Here the negative sign is in front of the radical, not under it. First evaluate the square root: $\sqrt{64} = 8$ because $8^2 = 64$ . Then apply the negation: $-\sqrt{64} = -8$ . Since $-8$ is an integer, $-\sqrt{64}$ is a real number.

The key distinction is the position of the negative sign: a negative radicand (inside the radical) produces a non-real result, while a negative sign placed before the radical simply yields the opposite of the principal square root, which remains real.

Updated 2026-04-21

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