Cube Root Calculator (∛x)
Calculate the cube root (third root) of any number. This calculator finds both real and complex cube roots, identifies perfect cubes, and shows detailed steps.
Tips:
• Positive numbers have one real cube root
• Negative numbers have one real and two complex cube roots
• Perfect cubes (e.g., 8, -27) give exact results
• Large numbers will use scientific notation
What is a Cube Root?
The cube root of a number is a value that, when cubed (raised to the power of 3), gives the original number. For example, 2 is the cube root of 8 because 2³ = 8.
If ∛x = n, then n³ = x
∛8 = 2 because 2³ = 8
∛(-27) = -3 because (-3)³ = -27
Example Calculations
Perfect Cubes
Positive Numbers
- ∛8 = 2 (since 2³ = 8)
- ∛27 = 3 (since 3³ = 27)
- ∛1000 = 10 (since 10³ = 1000)
Negative Numbers
- ∛(-8) = -2 (since (-2)³ = -8)
- ∛(-27) = -3 (since (-3)³ = -27)
- ∛(-1000) = -10 (since (-10)³ = -1000)
Complex Roots
Negative Numbers
Every negative number has one real and two complex cube roots:
For x = -8:
Three cube roots:
1. Real root: -2
2. Complex root: 2(cos(2π/3) + i·sin(2π/3))
3. Complex root: 2(cos(4π/3) + i·sin(4π/3))
Non-Perfect Cubes
Numbers that aren't perfect cubes have irrational cube roots:
∛10 ≈ 2.154434690031884
Verify:
(2.154434690031884)³ = 10
Properties of Cube Roots
- Every real number has exactly one real cube root
- Negative numbers have two additional complex cube roots
- The cube root of a negative number is negative
- Perfect cubes have rational cube roots
- The cube root function is strictly increasing