Cube Roots – Undoing a Cube

A cube root undoes cubing: ∛n asks “what number, multiplied by itself twice more, gives n?” Since 5 × 5 × 5 = 125, ∛125 = 5. Unlike square roots, cube roots exist for negative numbers too – ∛-8 = -2, since (-2) × (-2) × (-2) = -8 – because multiplying three negatives together gives a negative result, not a positive one.

Evaluating Cube Roots

∛n is the number that, cubed, equals n.

Evaluate ∛343.

7 × 7 × 7 = 343, so ∛343 = 7.

Solving Equations with Cube Roots

Solve x³ = 512 for x.

Take the cube root of both sides: x = ∛512 = 8.

Real-Life Application

Volume calculations: finding the side length of a cube from its volume.
Scaling models: working out a linear scale factor from a volume ratio.
Engineering: some material stress formulas involve cube roots.

Key Takeaways

∛n asks what number, cubed, gives n.
Unlike square roots, cube roots are defined for negative numbers too.
Perfect squares and perfect cubes, covered next, are the numbers that make these roots come out as whole numbers.

Practice: Cube Roots

Evaluate the Cube Root