Cube Root Calculator
Calculate the principal real cube root and all complex cube roots (∛x) with step-by-step solutions. Perfect for algebra, calculus, and understanding radical expressions.
Cube Roots Calculator
Solution:
Enter a number and click Calculate to see the cube root
What is a Cube Root?
The cube root of a number x is a value that, when multiplied by itself three times (cubed), equals x. It's written as ∛x or x^(1/3). For example, the cube root of 8 is 2 because 2 × 2 × 2 = 8. Unlike square roots, cube roots can be taken of negative numbers since a negative number cubed remains negative.
Mathematical Definition:
If ∛x = y, then y³ = x
Or: y × y × y = x
Perfect Cubes Reference
| Number | Cube (n³) | Cube Root (∛n³) |
|---|---|---|
| 1 | 1 | ∛1 = 1 |
| 2 | 8 | ∛8 = 2 |
| 3 | 27 | ∛27 = 3 |
| 4 | 64 | ∛64 = 4 |
| 5 | 125 | ∛125 = 5 |
| 6 | 216 | ∛216 = 6 |
| 7 | 343 | ∛343 = 7 |
| 8 | 512 | ∛512 = 8 |
| 9 | 729 | ∛729 = 9 |
| 10 | 1000 | ∛1000 = 10 |
Cube Root Properties
Basic Properties
- • ∛(a × b) = ∛a × ∛b
- • ∛(a / b) = ∛a / ∛b
- • ∛(a³) = a
- • (∛a)³ = a
- • ∛(-a) = -∛a
Special Cases
- • ∛0 = 0
- • ∛1 = 1
- • ∛(-1) = -1
- • ∛8 = 2
- • ∛27 = 3
Frequently Asked Questions
What is the difference between cube root and square root?
A square root (√x) finds a number that when multiplied by itself equals x, while a cube root (∛x) finds a number that when multiplied by itself three times equals x. Square roots of negative numbers are imaginary, but cube roots of negative numbers are real negative values.
Can you take the cube root of a negative number?
Yes! Unlike square roots, cube roots of negative numbers are real. For example, ∛(-8) = -2 because (-2)³ = -8. A negative number cubed is always negative, so its cube root is also negative.
How many cube roots does a number have?
Every non-zero number has exactly three cube roots: one real root and two complex conjugate roots. For positive numbers, the real root is positive. For negative numbers, the real root is negative. The complex roots involve imaginary numbers.
What is a perfect cube?
A perfect cube is a number that is the cube of an integer. Examples include 1 (1³), 8 (2³), 27 (3³), 64 (4³), etc. The cube root of a perfect cube is always a whole number. Perfect cubes can be positive or negative.
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