Cube Root Calculator

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Calculate cube roots, simplify radical expressions, and check for perfect cubes with detailed step-by-step solutions.

Calculator Mode

Enter any number (positive or negative)

About Cube Roots

The cube root of a number n is a value that, when cubed, gives n.

Example: ∛27 = 3 because 3³ = 27

Note: Cube roots of negative numbers are negative (∛-8 = -2)

Quick Reference

Common cube roots:

∛1 = 1∛8 = 2∛27 = 3∛64 = 4∛125 = 5∛216 = 6∛343 = 7∛512 = 8∛729 = 9∛1000 = 10

Properties of Cube Roots

  • ∛(a × b) = ∛a × ∛b (Product property)
  • ∛(a / b) = ∛a / ∛b (Quotient property)
  • ∛(a³) = a (Inverse property)
  • (∛a)³ = a (Cube of cube root)
  • ∛(-a) = -∛a (Negative numbers)
  • ∛0 = 0 (Zero property)
  • ∛1 = 1 (Identity property)

Simplification Rules

To simplify ∛n:

  1. Find prime factorization of n
  2. Group factors in sets of 3
  3. Extract each group as one factor
  4. Multiply extracted factors outside radical
  5. Keep remaining factors inside radical

Example: ∛54 = ∛(27 × 2) = ∛27 × ∛2 = 3∛2

Applications of Cube Roots

Geometry:

  • Finding cube side length from volume
  • 3D scaling calculations
  • Surface area to volume ratios
  • Sphere radius from volume

Science:

  • Cubic equations in physics
  • Density calculations
  • Growth rate models
  • Chemical concentration

Engineering:

  • Material stress analysis
  • Fluid dynamics
  • Structural design
  • Power calculations

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