Cube Root Calculator

Find the cube root of any real number. Includes perfect cube detection and nth root mode.

∛ Cube Root Calculator

Find the cube root of any real number, or switch to Nth Root mode for any degree.

-10001000
-10001000
th root
Cube Root (6 decimals)
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Rounded (4 decimals)
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Verification (result³)
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Perfect Cube?
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∛ What is a Cube Root?

The cube root of a number x is the value r such that r multiplied by itself three times equals x. Formally, if r³ = x then r is the cube root of x. In notation, this is written as the cube root symbol (a radical sign with a small 3 in the index), or equivalently as x raised to the power one-third. For example, the cube root of 27 is 3 because 3 × 3 × 3 = 27, and the cube root of 125 is 5 because 5 × 5 × 5 = 125.

Cube roots appear in many real-world contexts. In geometry, if you know the volume of a cube you find the side length by taking the cube root: a cube with volume 343 cubic centimetres has side length 7 cm. In physics, many scaling laws involve cube roots: the radius of a sphere is proportional to the cube root of its volume, and the orbital period of a planet relates to the cube root of its average distance from the sun via Kepler's third law. In statistics, the cube root transformation is used to reduce the skewness of right-skewed distributions before applying methods that assume symmetry.

An important property that distinguishes cube roots from square roots is that cube roots are defined for all real numbers, including negatives. The cube root of -8 is -2 because (-2) × (-2) × (-2) = -8. Square roots of negative numbers are not real, but cube roots of negatives always produce a real (negative) result. This makes the cube root function continuous across its entire domain with no restriction on the sign of x.

This calculator handles both Cube Root mode, where you enter any real number and get the exact cube root to six decimal places along with a perfect cube check, and Nth Root mode, where you can find any root degree from 2 through 10. The verification row confirms accuracy by cubing the result and showing that it matches the original input.

📐 Formula

³√x  =  x1/3
x = the number whose cube root you want (any real number)
r = x1/3 = cube root of x, satisfying r³ = x
nth root: ³√x generalises to the nth root: x1/n
Negative rule: ³√(−x) = −³√x for all positive x
Example: ³√512 = 5121/3 = 8, because 8 × 8 × 8 = 512
Example: ³√(−64) = −4, because (−4)³ = −64

📖 How to Use This Calculator

Steps

1
Enter the number: type any real number into the input field. Negative numbers are allowed in Cube Root mode.
2
Click Calculate: the calculator returns the cube root to 6 decimal places and checks whether the input is a perfect cube.
3
Read the verification: the Verification row shows the cube root raised to the third power. It should equal your original input, confirming the result is correct.
4
Switch to Nth Root for other roots: select the Nth Root mode, enter a base number and a root degree from 2 to 10, then click Calculate to find any root.

💡 Example Calculations

Example 1: Cube root of 512

Finding the cube root of a perfect cube

1
x = 512. Apply the formula: r = 5121/3.
2
8 × 8 × 8 = 512. So r = 8.
3
Verification: 8³ = 512. Result is a perfect cube.
Cube root of 512 = 8 (perfect cube)
Try this example →

Example 2: Cube root of 2 (irrational)

Cube root of a non-perfect cube

1
x = 2. r = 21/3. This is irrational (no exact fraction).
2
Calculator result: r ≈ 1.259921. Rounded to 4 decimals: 1.2599.
3
Verify: 1.259921³ ≈ 2.000000. Confirmed.
Cube root of 2 ≈ 1.2599 (irrational)
Try this example →

Example 3: Cube root of −125

Negative input, negative real cube root

1
x = −125. Cube roots of negatives are negative real numbers.
2
(−5) × (−5) × (−5) = −125. So r = −5.
3
Verification: (−5)³ = −125. Perfect cube.
Cube root of −125 = −5 (perfect cube)
Try this example →

Example 4: 5th root of 243 in Nth Root mode

Using the Nth Root mode for degree 5

1
Switch to Nth Root mode. Set base = 243, degree = 5.
2
r = 2431/5. 3&sup5; = 243, so r = 3.
3
Verify: 3&sup5; = 243. Integer result confirmed.
5th root of 243 = 3
Try this example →

❓ Frequently Asked Questions

What is the cube root of a number?+
The cube root of a number x is the value r such that r cubed equals x. It is written with a radical sign and the index 3, or equivalently as x to the power one-third. For example, the cube root of 27 is 3 because 3 times 3 times 3 equals 27. The cube root function is the inverse of the cube function.
How do you find the cube root without a calculator?+
For perfect cubes, memorise the list: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. For other numbers, use the successive approximation method: guess a starting value, cube it, compare with the target, and adjust the guess. The Newton-Raphson method refines this: the next guess is (2 times current guess plus x divided by current guess squared) divided by 3.
Is the cube root of a negative number a real number?+
Yes. Unlike even roots, odd roots of negative numbers are real. The cube root of -8 is -2 because (-2) times (-2) times (-2) equals -8. In general, the cube root of negative x equals negative the cube root of positive x. This means the cube root function is defined for all real numbers with no domain restriction.
What are the first ten perfect cubes?+
The first ten perfect cubes are 1 (cube root 1), 8 (cube root 2), 27 (cube root 3), 64 (cube root 4), 125 (cube root 5), 216 (cube root 6), 343 (cube root 7), 512 (cube root 8), 729 (cube root 9), and 1000 (cube root 10). A perfect cube is any integer that is the cube of another integer. This calculator shows a Yes or No indicator for whether the input is a perfect cube.
What is the difference between a cube root and a square root?+
The square root finds a value r where r squared equals x; the cube root finds r where r cubed equals x. Key differences: square roots of negative numbers are not real, but cube roots of negatives always are. Every positive number has two square roots (plus and minus) but exactly one real cube root. The square root function is only defined for non-negative inputs; the cube root is defined for all real numbers.
How is the cube root related to exponents?+
The cube root of x equals x to the power one-third. This is the general rule: the nth root of x equals x to the power one over n. So the cube root follows directly from exponent rules. You can compute it on any scientific calculator using the exponent key: enter the number, raise it to the power 1/3 or 0.3333, and the result is the cube root.
What real-world problems use cube roots?+
Cube roots appear in volume problems (a cube with volume V has side length equal to the cube root of V), in physics (the radius of a sphere with volume V is the cube root of 3V divided by 4 pi), in astronomy (Kepler's third law links orbital period to the cube root of orbital radius), in statistics (cube root transformation reduces skewness), and in chemistry (unit cell edge length from crystal volume).
What is the cube root of 0 and 1?+
The cube root of 0 is 0 and the cube root of 1 is 1. These are fixed points of the cube root function: they equal their own cube root. The cube root of 0 is trivially 0 because 0 cubed is 0. The cube root of 1 is 1 because 1 cubed is 1. No other non-negative real number equals its own cube root.
How do you simplify the cube root of a fraction?+
The cube root of a fraction a over b equals the cube root of a divided by the cube root of b. For example, the cube root of 8 over 27 equals the cube root of 8 divided by the cube root of 27, which equals 2 divided by 3. This works because the nth root distributes over multiplication and division. Simplify by finding perfect cube factors in both numerator and denominator.
What is the cube root of a cube? Is the cube root of x cubed equal to x?+
The cube root of x cubed equals x for all real numbers. This is because the cube root and the cube operation are inverse functions: applying one undoes the other. Formally, (x^(1/3))^3 equals x and (x^3)^(1/3) equals x. There is no restriction on sign: the cube root of (-5)^3 = the cube root of -125 = -5, confirming the identity holds for negatives too.
What is the cube root of a large number like 1 million?+
The cube root of 1,000,000 is 100, because 100 times 100 times 100 equals 1,000,000. A quick shortcut for powers of 10: the cube root of 10^n equals 10^(n/3). For 10^6, that is 10^2 = 100. For 10^9 (one billion), the cube root is 10^3 = 1,000. This pattern makes mental estimation fast for numbers that are exact powers of 10.
Tags: Cube Root Calculator Cube Root Nth Root Calculator Perfect Cube Cbrt Calculator Formula Free Online

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What is the cube root of a number?

The cube root of a number x is the value r such that r multiplied by itself three times equals x. Written as the cube root symbol (3rd radical) of x, or equivalently x raised to the power one-third. For example the cube root of 27 is 3 because 3 x 3 x 3 equals 27.

How do you find the cube root of a perfect cube?

For a perfect cube, the cube root is an exact integer. Memorise or look up the cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. Recognise the number in that list and the cube root is its position. For 512, the position is 8 because 8 cubed is 512.

Can the cube root of a negative number be negative?

Yes. Unlike even roots (square, fourth, sixth), odd roots always exist for any real number. The cube root of a negative number is negative. The cube root of -8 is -2, the cube root of -27 is -3, and the cube root of -125 is -5.

What is the formula for a cube root?

The cube root of x equals x raised to the power one-third. In exponent notation: x^(1/3). This follows the general rule that the nth root of x equals x^(1/n). You can also write it using the radical sign with the small 3 in the index position.

Is cube root the same as dividing by 3?

No. Cube root and dividing by 3 are completely different operations. The cube root of 27 is 3 because 3 cubed equals 27. Dividing 27 by 3 gives 9. They only coincide for special values by coincidence. Cube root is the inverse of cubing; dividing by 3 is the inverse of multiplying by 3.

What are perfect cubes up to 1000?

The perfect cubes up to 1000 are: 1 (1 cubed), 8 (2 cubed), 27 (3 cubed), 64 (4 cubed), 125 (5 cubed), 216 (6 cubed), 343 (7 cubed), 512 (8 cubed), 729 (9 cubed), and 1000 (10 cubed). These are the only positive integers up to 1000 whose cube roots are whole numbers.

How is the cube root used in real life?

Cube roots appear in volume calculations: if a cube has volume V, its side length is the cube root of V. They appear in physics for scaling laws (the cube root of mass relates to linear size in many organisms), in statistics for transforming right-skewed data, and in finance for computing compound growth rates over three periods.

What is the cube root of 0 and 1?

The cube root of 0 is 0 and the cube root of 1 is 1. These are the only non-negative numbers that equal their own cube root. This follows from the exponent rule: 0 raised to any positive power is 0, and 1 raised to any power is 1.

📌 Quick Tips

💡The cube root of a negative number is negative. For example, the cube root of -8 is -2 because (-2) x (-2) x (-2) = -8.
💡A perfect cube is an integer whose cube root is also an integer, such as 8, 27, 64, 125, and 216.
💡To verify a cube root manually, multiply the result by itself three times. If you get back the original number, the answer is correct.
💡The cube root of 0 is 0 and the cube root of 1 is 1. These are the only numbers that equal their own cube root.
💡Unlike square roots, cube roots of negative numbers are real numbers, not complex. The cube root of -27 is -3.