Cube Root of −5

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What is the cube root of −5?

Answer

∛(−5) =
−∛5 ≈ −1.71

Real cube root: the only number whose cube is −5.

≈ — rounded to 4 decimal places. −5 is not the cube of a rational number, so ∛(−5) is irrational: its decimal expansion never ends and never repeats.

A negative number has a negative cube root: (−∛5)³ = −5. There is no real square root of −5.

Sign
∛(−5) = −∛5, because (−a)³ = −a³
Prime factors
5 is a prime number: no triple of equal factors.
Between cubes
1³ = 1 < 5 < 8 = 2³, so 1 < ∛5 < 2 and ∛5 ≈ 1.71

Related Calculations

See Also

Recalculate with different values ↑

Cube roots of nearby values

N∛N
−15−∛15 ≈ −2.4662
−8−2
−7−∛7 ≈ −1.9129
−5−∛5 ≈ −1.71
−3−∛3 ≈ −1.4422
−2−∛2 ≈ −1.2599
−1−1

FAQ

What is the cube root of −5?

∛(−5) = −∛5 ≈ −1.71

Why does the cube root of −5 exist but the square root does not?

A cube keeps the sign: (−∛5)³ = −5, so ∛(−5) = −∛5. The square of any real number is never negative, so no real number squared gives −5.

Is ∛(−5) a rational number?

No: −5 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 ∛(−5) is irrational; its decimal never ends and never repeats: ∛(−5) ≈ −1.71.

See Also