Cube Root Calculator
Calculate Cube Roots
Find principal cube roots, simplify radicals, and get exact or approximate solutions for positive and negative numbers.
Cube Root Result
∛27 = 3
Number Properties:
Radical Simplification:
Cube Root Magnitude:
The cube root of a number is the value that, when multiplied by itself three times, gives the original number.
What is a Cube Root?
Cube Root of a number is a value that, when multiplied by itself three times (cubed), gives the original number. For example, the cube root of 27 is 3 because 3 × 3 × 3 = 27. Unlike square roots, cube roots can be negative since negative numbers cubed give negative results.
Cube Root Properties
Real Cube Root
Works for all reals
Positive & negative
Perfect Cubes
Integer results
Exact values
Negative Numbers
Negative results
Real solutions
Radical Form
Simplification rules
Exact expressions
Cube Root Rules
1. Perfect Cube Definition
A perfect cube is a number that has an integer as its cube root:
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, ...
2. Radical Simplification
Cube roots can be simplified using the product property:
∛(a × b) = ∛a × ∛b
3. Cube Root of Fractions
The cube root of a fraction equals the cube root of numerator divided by cube root of denominator:
∛(a/b) = ∛a / ∛b
Real-World Applications
Geometry & Volume Calculations
- Cube volume: Finding side length from volume of cubes
- 3D modeling: Scaling objects proportionally
- Container design: Calculating dimensions from capacity
- Architecture: Space planning and volume optimization
Physics & Engineering
- Density calculations: Finding dimensions from mass and density
- Thermodynamics: Volume relationships in gas laws
- Materials science: Crystal structure analysis
- Fluid dynamics: Flow rate and pipe diameter calculations
Mathematics & Algebra
- Equation solving: Cubic equation solutions
- Function analysis: Understanding cubic functions
- Calculus: Volume integration problems
- Number theory: Perfect cube properties
Computer Graphics & Gaming
- 3D rendering: Object scaling and transformations
- Game development: Volume-based calculations
- Animation: Proportional scaling of 3D models
- VR/AR: Spatial calculations and measurements
Common Cube Root Examples
| Number | Cube Root | Type | Simplified Radical |
|---|---|---|---|
| 8 | 2 | Perfect Cube | 2 |
| 27 | 3 | Perfect Cube | 3 |
| -64 | -4 | Perfect Cube | -4 |
| 54 | ≈ 3.780 | Non-Perfect | 3∛2 |
Perfect Cubes Reference
| Number | Cube | Cube Root | Number | Cube | Cube Root |
|---|---|---|---|---|---|
| 1 | 1 | 1 | 6 | 216 | 6 |
| 2 | 8 | 2 | 7 | 343 | 7 |
| 3 | 27 | 3 | 8 | 512 | 8 |
| 4 | 64 | 4 | 9 | 729 | 9 |
| 5 | 125 | 5 | 10 | 1000 | 10 |
Step-by-Step Calculation Process
Example 1: Perfect Cube (∛216)
- Identify the number: 216
- Check if it's a perfect cube: 6 × 6 × 6 = 216
- Result: ∛216 = 6
- Verification: 6 × 6 × 6 = 216 ✓
Example 2: Non-Perfect Cube (∛54)
- Identify the number: 54
- Find perfect cube factors: 54 = 27 × 2
- Apply radical rule: ∛54 = ∛(27 × 2) = ∛27 × ∛2
- Simplify: 3 × ∛2
- Decimal approximation: 3 × 1.260 = 3.780
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Frequently Asked Questions (FAQs)
Q: Can cube roots be negative?
A: Yes! Unlike square roots, cube roots can be negative. For example, ∛(-8) = -2 because (-2) × (-2) × (-2) = -8.
Q: What is the cube root of 0?
A: The cube root of 0 is 0, since 0 × 0 × 0 = 0.
Q: How do you simplify cube roots?
A: Factor the number under the radical into perfect cubes and other factors, then take the cube root of the perfect cubes outside the radical.
Q: What's the difference between cube roots and square roots?
A: Square roots find what number squared equals the original, while cube roots find what number cubed equals the original. Cube roots work for negative numbers, square roots don't (in real numbers).
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