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    Cube Root of −108

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    Cube root of

    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

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    https://calculat.io/en/number/cube-root/-108Copy
    <a href="https://calculat.io/en/number/cube-root/-108">Cube Root of −108 (∛(−108)): Exact Value and Decimal</a>Copy
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    Cube Root of −108 (∛(−108)): Exact Value and Decimal. ∛(−108) = −3∛4 ≈ −4.7622. 108 = 3³ × 4, so 3 comes out of the root. The minus comes out of the root. Prime factors in threes and a check.
    https://calculat.io/en/number/cube-root/-108/generated.jpgCopy

    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
    Recalculate with different values ↑

    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