Cube Root of −106875

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What is the cube root of −106,875?

Answer

∛(−106,875) =
−5∛855 ≈ −47.4561

Real cube root: the only number whose cube is −106,875.

≈ — rounded to 4 decimal places. −106,875 is not the cube of a rational number, so ∛(−106,875) is irrational: its decimal expansion never ends and never repeats.

A negative number has a negative cube root: (−5∛855)³ = −106,875. There is no real square root of −106,875.

Sign
∛(−106,875) = −∛106,875, because (−a)³ = −a³
Prime factors
106,875 = 3 × 3 × (5 × 5 × 5) × 5 × 19 = 3² × 5⁴ × 19
Take out of the root
∛106,875 = ∛(5³ × 855) = 5∛855
Between cubes
47³ = 103,823 < 106,875 < 110,592 = 48³, so 47 < ∛106,875 < 48 and ∛106,875 ≈ 47.4561

See Also

Recalculate with different values ↑

Cube roots of nearby values

N∛N
−106,875−5∛855 ≈ −47.4561
−8,000−20
−6,859−19
−5,832−18
−4,913−17
−4,096−16
−3,375−15

FAQ

What is the cube root of −106,875?

∛(−106,875) = −5∛855 ≈ −47.4561

Why does the cube root of −106,875 exist but the square root does not?

A cube keeps the sign: (−5∛855)³ = −106,875, so ∛(−106,875) = −5∛855. The square of any real number is never negative, so no real number squared gives −106,875.

Is ∛(−106,875) a rational number?

No: −106,875 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 ∛(−106,875) is irrational; its decimal never ends and never repeats: ∛(−106,875) ≈ −47.4561.

See Also