Cube Root of −729

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What is the cube root of −729?

Answer

∛(−729) =
−9

Real cube root: the only number whose cube is −729.

A negative number has a negative cube root: (−9)³ = −729. There is no real square root of −729.

Sign
∛(−729) = −∛729, because (−a)³ = −a³
Prime factors
729 = (3 × 3 × 3) × (3 × 3 × 3) = 3⁶
Take out of the root
∛729 = ∛((3³)²) = 3² = 9
Check
9 × 9 × 9 = 729 ✓

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Cube roots of nearby values

N∛N
−1,728−12
−1,331−11
−1,000−10
−729−9
−512−8
−343−7
−216−6

FAQ

What is the cube root of −729?

∛(−729) = −9

Why does the cube root of −729 exist but the square root does not?

A cube keeps the sign: (−9)³ = −729, so ∛(−729) = −9. The square of any real number is never negative, so no real number squared gives −729.

Is ∛(−729) a rational number?

Yes: ∛(−729) = −9, a whole number.

See Also