Calculate the principal real cube root of positive and negative numbers. Identifies perfect cubes, nearest integer roots, and complex conjugate roots.
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The Cube Root Calculator computes the real principal cube root (∛x) of any positive or negative real number. Unlike square roots, real cube roots exist for all negative numbers because an odd power of a negative number is negative.
y = \sqrt[3]{x} \iff y^3 = x, \quad \sqrt[3]{-x} = -\sqrt[3]{x}Every non-zero real number has three cube roots: one real principal root and two complex conjugate roots spaced at 120° angles in the complex plane.
Scenario: Calculating the cube root of -64 and 50
For x = -64: (-4) × (-4) × (-4) = -64 → ∛(-64) = -4 (Perfect cube).
For x = 50: 3³ = 27 < 50 < 4³ = 64 → ∛50 ≈ 3.684031.
Yes! Because (-a)³ = -a³, every negative real number has an exact real negative cube root. For example, ∛(-8) = -2.
1 (1³), 8 (2³), 27 (3³), 64 (4³), 125 (5³), 216 (6³), 343 (7³), 512 (8³), 729 (9³), and 1,000 (10³).
Factor the number into prime factors. For every group of three identical factors (e.g., 2 × 2 × 2), pull one factor outside the radical sign.
If r is the real principal cube root, the two complex conjugate roots are r · (-0.5 + i√3/2) and r · (-0.5 - i√3/2).
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