Cube Root of −81

Cube Root Calculator
Cube root of

What is the cube root of −81?

Answer

∛(−81) =
−3∛3 ≈ −4.3267

Real cube root: the only number whose cube is −81.

≈ — rounded to 4 decimal places. −81 is not the cube of a rational number, so ∛(−81) is irrational: its decimal expansion never ends and never repeats.

A negative number has a negative cube root: (−3∛3)³ = −81. There is no real square root of −81.

Sign
∛(−81) = −∛81, because (−a)³ = −a³
Prime factors
81 = (3 × 3 × 3) × 3 = 3⁴
Take out of the root
∛81 = ∛(3³ × 3) = 3∛3
Between cubes
4³ = 64 < 81 < 125 = 5³, so 4 < ∛81 < 5 and ∛81 ≈ 4.3267

Related Calculations

See Also

Recalculate with different values ↑

Cube roots of nearby values

N∛N
−216−6
−125−5
−108−3∛4 ≈ −4.7622
−81−3∛3 ≈ −4.3267
−64−4
−54−3∛2 ≈ −3.7798
−49−∛49 ≈ −3.6593

FAQ

What is the cube root of −81?

∛(−81) = −3∛3 ≈ −4.3267

Why does the cube root of −81 exist but the square root does not?

A cube keeps the sign: (−3∛3)³ = −81, so ∛(−81) = −3∛3. The square of any real number is never negative, so no real number squared gives −81.

Is ∛(−81) a rational number?

No: −81 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 ∛(−81) is irrational; its decimal never ends and never repeats: ∛(−81) ≈ −4.3267.

See Also