Arithmetic
How To Calculate A Cube Root
A cube root asks which number, multiplied by itself twice more, produces the value you started with. It is the exact inverse of cubing, and unlike a square root it has a single real answer for every real input, negative numbers included.
Quick Answer
cube root of x = x^(1/3), with the sign of x carried separately
- x
- The number whose cube root you want
- |x|
- The absolute value, which strips the sign
- sign(x)
- Either 1 or -1, restoring the original sign
- x^(1/3)
- The cube root, written with a fractional exponent
Take the absolute value, raise it to the power of one third, then reapply the sign of the original number. The cube root of 27 is exactly 3, the cube root of -27 is exactly -3, and the cube root of 64 evaluates to 3.9999999999999996 before it is rounded to 4.
What Is A Cube Root?
Cubing a number means multiplying it by itself twice more: 4 cubed is 4 x 4 x 4, which is 64. The cube root runs that operation backwards. It answers the question of which number was cubed, so the cube root of 64 is 4.
Fractional exponents are the standard way to write roots. An exponent of 1/2 is a square root, an exponent of 1/3 is a cube root, and an exponent of 1/n is the nth root. That single piece of notation covers every root, and it is why calculators usually label the key with an x raised to a fractional power rather than a dedicated cube root button.
Cube roots differ from square roots in one important way: every real number has exactly one real cube root. Squaring any real number gives a positive or zero result, so a negative number has no real square root and the square root symbol usually means only the positive one when it does exist. Cubing preserves the sign, so -3 cubed is -27 and the cube root of -27 is unambiguously -3.
Because the sign passes through cleanly, you can compute the cube root of a negative number in two steps: take the cube root of the absolute value, then negate. Trying to feed a negative number straight into a fractional exponent is the usual source of errors, since many implementations return a complex result instead of a real one.
The cube root of a product is the product of the cube roots, and the same holds for quotients. That is useful for mental maths: the cube root of 8,000 is the cube root of 8 times the cube root of 1,000, which is 2 x 10, or 20. Splitting a number into a tidy part and a power of ten turns a hard root into an easy one.
Perfect cubes are worth memorising up to about 20 cubed, because they turn an estimate into an exact answer. One, 8, 27, 64, 125, 216, 343, 512, 729, 1,000, 1,331, 1,728, 2,197, 2,744, 3,375, 4,096, 4,913, 5,832, 6,859 and 8,000 cover most of what comes up in practice.
Cube roots appear wherever a quantity is defined per unit of volume rather than per unit of length. Density times volume gives mass, so working backwards from a mass to a linear dimension needs a cube root. Scaling laws in biology and engineering often use cube roots for the same reason.
Floating point arithmetic rarely returns a clean integer. The IEEE 754 double used by essentially every programming language stores the cube root of 64 as 3.9999999999999996 rather than 4, because one third cannot be represented exactly in binary and the rounding error accumulates in the power function.
Rounding fixes the display but hides the underlying value. Rounding the cube root of 64 to six decimal places gives 4, and cubing that rounded value returns exactly 64. Keeping several decimal places in intermediate steps and rounding only the final answer is the habit that stops these errors compounding.
A quick sanity check is to cube whatever answer you produced. If the cube root of 64 is claimed to be 4, cubing 4 must return 64. If it does not, the root is wrong, and the gap between your cubed answer and the original number tells you how far off you are.
Formula
cube root of x = sign(x) * |x|^(1/3)
Formula page →The absolute value keeps the power function in positive territory, and multiplying by the sign restores direction.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| x | Input | any unit | The number whose cube root you want. |
| |x| | Absolute value | same unit | The input with its sign removed. |
| sign(x) | Sign | unitless | One for positive input, minus one for negative input. |
| 1/3 | Fractional exponent | unitless | The exponent that inverts cubing. |
check = (cube root of x) x (cube root of x) x (cube root of x)
Formula page →Cubing the result must return the original number. This is how you confirm a cube root by hand.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| r | Cube root | same unit | The answer you produced. |
| check | Verification | same unit | The original number, recovered by cubing. |
| r^3 | Cube of r | same unit | Three copies of r multiplied together. |
How To Calculate A Cube Root
- 1
Take the absolute value
Strip the sign so the power function only ever sees a positive number. The cube root of 27 and the cube root of -27 share the same magnitude, which is 27.
- 2
Raise it to the power of one third
An exponent of one third inverts cubing. Applied to 27 this gives approximately 3, and applied to 64 it gives 3.9999999999999996 in double precision floating point.
- 3
Reapply the sign
Multiply by one if the input was positive and by minus one if it was negative. Cubing a negative number stays negative, so the root must too.
- 4
Cube the answer to check it
Multiply your result by itself twice more. You should land back on the original number, or close enough to it that the difference is explained by rounding.
- 5
Round only the final answer
Keep full precision through intermediate steps and round once at the end. Rounding early, then cubing the rounded value, will not return the original number for non-perfect cubes.
Examples
Example 1: The cube root of 27
- Number
- 27
| Step | Calculation | Result |
|---|---|---|
| Absolute value | abs(27) | 27 |
| Sign | 27 / 27 | 1 |
| Cube root | 1 * 27^(1/3) | 3 |
| Cube of the result | 3 x 3 x 3 | 27 |
| Gap to the nearest perfect cube | abs(3 x 3 x 3 - 27) | 0 |
Result: The cube root of 27 is exactly 3. Cubing that result returns 27, and the nearest whole root is also 3, which sits 0 away from the nearest perfect cube because 27 is a perfect cube.
Example 2: The cube root of 64, where floating point bites
- Number
- 64
| Step | Calculation | Result |
|---|---|---|
| Absolute value | abs(64) | 64 |
| Sign | 64 / 64 | 1 |
| Cube root before rounding | 1 * 64^(1/3) | 3.9999999999999996 |
| Rounded to six decimals | round(3.9999999999999996, 6) | 4 |
| Cube of the rounded root | 4 x 4 x 4 | 64 |
Result: The cube root of 64 evaluates to 3.9999999999999996 in double precision, which rounds to 4 at six decimal places, and cubing 4 returns 64 exactly, so the exact answer is 4.
Example 3: The cube root of a negative number
- Number
- -27
| Step | Calculation | Result |
|---|---|---|
| Absolute value | abs(-27) | 27 |
| Sign | -27 / 27 | -1 |
| Cube root | -1 * 27^(1/3) | -3 |
| Cube of the result | -3 x -3 x -3 | -27 |
| Gap to the nearest perfect cube | abs(-3 x -3 x -3 - 27) | 54 |
Result: The cube root of -27 is -3, because -3 cubed is -27. Negative inputs have a single real cube root, so nothing complex is involved, and the gap between the cube of that root and the absolute value of the input works out at 54, the full distance.
Calculator
Cube root
3
- Nearest whole cube root
- 3
- Cube of the rounded root
- 27
- Gap to the nearest perfect cube
- 0
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
Prefer a full-width tool? Open the A Cube Root calculator page.
Common Mistakes
Feeding a negative number straight into a fractional exponent
Many calculators and spreadsheet functions return an error or a complex number when asked to raise a negative value to the power of one third. Splitting the job into an absolute value and a sign avoids that entirely.
Confusing the cube root with the square root
The square root of -27 does not exist as a real number, while the cube root of -27 is exactly -3. Mixing the two up produces an error message where an answer exists.
Cubing the root instead of checking it
Cubing the answer is a check, not a separate step in the calculation. If you cube the cube root you simply recover the input, which confirms the work rather than advancing it.
Rounding before the final answer
Rounding 3.9999999999999996 to 4 and then cubing gives 64, but rounding 9.999999999999998 to 10 and cubing gives 1,000 where the true cube of the rounded root would not. Keep full precision while working and round once at the end.
Splitting the number at the wrong place
The cube root of 8,000 is the cube root of 8 times the cube root of 1,000, which is 2 x 10. Cutting 8,000 into 80 and 100 instead gives cube roots of 4.31 and 4.64, which do not multiply back to 20.
Assuming a cube root can never be negative
Every real number has exactly one real cube root, and it carries the sign of the input. The cube root of -64 is -4, not undefined.
FAQ
What is the cube root of 27?
The cube root of 27 is 3, because 3 cubed is 27. It is an exact answer, not an approximation, and it needs no rounding.
What is the cube root of 64?
The cube root of 64 is 4. Most software evaluates it as 3.9999999999999996 first and returns 4 once the result is rounded, because one third cannot be stored exactly in binary floating point.
Does a negative number have a cube root?
Yes, exactly one, and it is negative. The cube root of -27 is -3, because -3 cubed is -27. This is the main way cube roots differ from square roots.
How do I write a cube root as an exponent?
Raise the number to the power of one third. A cube root is x^(1/3), a square root is x^(1/2), and the nth root is x^(1/n).
How do I estimate a cube root by hand?
Find the two nearest perfect cubes and interpolate. Between 3 cubed, which is 27, and 4 cubed, which is 64, the cube root of 50 sits closer to 4 than to 3, and a single round of trial and error lands within a couple of percent.
Why is there only one cube root symbol but three cube roots of a complex number?
In real arithmetic a number has one real cube root. In complex arithmetic the equation r^3 = x has three distinct solutions, and the principal one is the real value that the cube root symbol reports.
References
- [1]Wikipedia, Cube root — https://en.wikipedia.org/wiki/Cube_root
- [2]Wikipedia, Nth root — https://en.wikipedia.org/wiki/Nth_root
- [3]Wikipedia, IEEE 754 double-precision floating-point format — https://en.wikipedia.org/wiki/Double-precision_floating-point_format