Cube Root of −1000

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What is the cube root of −1,000?

Answer

∛(−1,000) =
−10

Real cube root: the only number whose cube is −1,000.

A negative number has a negative cube root: (−10)³ = −1,000. There is no real square root of −1,000.

Sign
∛(−1,000) = −∛1,000, because (−a)³ = −a³
Prime factors
1,000 = (2 × 2 × 2) × (5 × 5 × 5) = 2³ × 5³
Take out of the root
∛1,000 = ∛(2³ × 5³) = 2 × 5 = 10
Check
10 × 10 × 10 = 1,000 ✓

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FAQ

What is the cube root of −1,000?

∛(−1,000) = −10

Why does the cube root of −1,000 exist but the square root does not?

A cube keeps the sign: (−10)³ = −1,000, so ∛(−1,000) = −10. The square of any real number is never negative, so no real number squared gives −1,000.

Is ∛(−1,000) a rational number?

Yes: ∛(−1,000) = −10, a whole number.

See Also