Cube Root of −108

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What is the cube root of −108?

Answer

∛(−108) =
−3∛4 ≈ −4.7622

Real cube root: the only number whose cube is −108.

≈ — rounded to 4 decimal places. −108 is not the cube of a rational number, so ∛(−108) is irrational: its decimal expansion never ends and never repeats.

A negative number has a negative cube root: (−3∛4)³ = −108. There is no real square root of −108.

Sign
∛(−108) = −∛108, because (−a)³ = −a³
Prime factors
108 = 2 × 2 × (3 × 3 × 3) = 2² × 3³
Take out of the root
∛108 = ∛(3³ × 4) = 3∛4
Between cubes
4³ = 64 < 108 < 125 = 5³, so 4 < ∛108 < 5 and ∛108 ≈ 4.7622

Related Calculations

See Also

Recalculate with different values ↑

Cube roots of nearby values

N∛N
−343−7
−216−6
−125−5
−108−3∛4 ≈ −4.7622
−81−3∛3 ≈ −4.3267
−64−4
−54−3∛2 ≈ −3.7798

FAQ

What is the cube root of −108?

∛(−108) = −3∛4 ≈ −4.7622

Why does the cube root of −108 exist but the square root does not?

A cube keeps the sign: (−3∛4)³ = −108, so ∛(−108) = −3∛4. The square of any real number is never negative, so no real number squared gives −108.

Is ∛(−108) a rational number?

No: −108 is not the cube of a rational number, and the cube root of a fraction p/q in lowest terms is rational only when p and q are both cubes. So ∛(−108) is irrational; its decimal never ends and never repeats: ∛(−108) ≈ −4.7622.

See Also