Cube Root of −512

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Cube root of

What is the cube root of −512?

Answer

∛(−512) =
−8

Real cube root: the only number whose cube is −512.

A negative number has a negative cube root: (−8)³ = −512. There is no real square root of −512.

Sign
∛(−512) = −∛512, because (−a)³ = −a³
Prime factors
512 = (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2) = 2⁹
Take out of the root
∛512 = ∛((2³)³) = 2³ = 8
Check
8 × 8 × 8 = 512 ✓

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Cube roots of nearby values

N∛N
−1,331−11
−1,000−10
−729−9
−512−8
−343−7
−216−6
−125−5

FAQ

What is the cube root of −512?

∛(−512) = −8

Why does the cube root of −512 exist but the square root does not?

A cube keeps the sign: (−8)³ = −512, so ∛(−512) = −8. The square of any real number is never negative, so no real number squared gives −512.

Is ∛(−512) a rational number?

Yes: ∛(−512) = −8, a whole number.

See Also