Sqrt of -32

The principal square root of −32 is 4√2 i in the complex numbers. There is no real number whose square is negative thirty-two. The number i is defined so that i² = −1.

Meaning and context

Factor 32 as 16 × 2. The magnitude of the square root is therefore √32 = 4√2. Including the imaginary unit gives √(−32) = 4√2 i under the principal complex-square-root convention.

Examples and usage

Check by squaring: (4√2 i)² = 16 × 2 × (−1) = −32. If the question instead asks you to solve x² = −32, both 4√2 i and −4√2 i are solutions.

What to check

Distinguish the principal square root from the two solutions of a quadratic equation. Also do not apply the real-number rule √a√b = √(ab) indiscriminately to negative inputs; complex square roots require care with those identities.

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