Cube Root of −32

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What is the cube root of −32?

Answer

∛(−32) =
−2∛4 ≈ −3.1748

Real cube root: the only number whose cube is −32.

≈ — rounded to 4 decimal places. −32 is not the cube of a rational number, so ∛(−32) is irrational: its decimal expansion never ends and never repeats.

A negative number has a negative cube root: (−2∛4)³ = −32. There is no real square root of −32.

Sign
∛(−32) = −∛32, because (−a)³ = −a³
Prime factors
32 = (2 × 2 × 2) × 2 × 2 = 2⁵
Take out of the root
∛32 = ∛(2³ × 4) = 2∛4
Between cubes
3³ = 27 < 32 < 64 = 4³, so 3 < ∛32 < 4 and ∛32 ≈ 3.1748

Related Calculations

See Also

Recalculate with different values ↑

Cube roots of nearby values

N∛N
−64−4
−54−3∛2 ≈ −3.7798
−49−∛49 ≈ −3.6593
−32−2∛4 ≈ −3.1748
−27−3
−25−∛25 ≈ −2.924
−21−∛21 ≈ −2.7589

FAQ

What is the cube root of −32?

∛(−32) = −2∛4 ≈ −3.1748

Why does the cube root of −32 exist but the square root does not?

A cube keeps the sign: (−2∛4)³ = −32, so ∛(−32) = −2∛4. The square of any real number is never negative, so no real number squared gives −32.

Is ∛(−32) a rational number?

No: −32 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 ∛(−32) is irrational; its decimal never ends and never repeats: ∛(−32) ≈ −3.1748.

See Also