Cube Root of −8000

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What is the cube root of −8,000?

Answer

∛(−8,000) =
−20

Real cube root: the only number whose cube is −8,000.

A negative number has a negative cube root: (−20)³ = −8,000. There is no real square root of −8,000.

Sign
∛(−8,000) = −∛8,000, because (−a)³ = −a³
Prime factors
8,000 = (2 × 2 × 2) × (2 × 2 × 2) × (5 × 5 × 5) = 2⁶ × 5³
Take out of the root
∛8,000 = ∛((2³)² × 5³) = 2² × 5 = 20
Check
20 × 20 × 20 = 8,000 ✓

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FAQ

What is the cube root of −8,000?

∛(−8,000) = −20

Why does the cube root of −8,000 exist but the square root does not?

A cube keeps the sign: (−20)³ = −8,000, so ∛(−8,000) = −20. The square of any real number is never negative, so no real number squared gives −8,000.

Is ∛(−8,000) a rational number?

Yes: ∛(−8,000) = −20, a whole number.

See Also