Cube Root Calculator

A cube root calculator answers one question: which number, multiplied by itself three times, produces the number you typed? For a perfect cube the answer is exact, so the cubed root of 125 is 5. For everything else it is irrational: the cube root of 250 is 6.29960525, or exactly 5 times the cube root of 2.

Unlike a square root, a cube root never runs out of answers. Every real number, positive, negative or zero, has exactly one real cube root, and that root carries the same sign as the input. That single fact is what makes cube roots so useful for volumes, growth rates and scaling problems, where a negative or a fractional input is completely normal.

How a cube root is calculated

The cube root is the inverse of cubing. If you cube a number you multiply it by itself three times; the cube root undoes that. Formally, the cube root of x is the value y that satisfies:

The second form is the practical one. A root is nothing more than a fractional exponent, so the cube root of x equals x raised to the power one third. The general nth root follows the same rule:

That is why every scientific calculator can produce a cube root even without a dedicated button, and why spreadsheet users write =250^(1/3). The calculator above uses the same exponent, then applies one Newton refinement step so the last digits are correct rather than drifting from the rounded exponent.

Worked example: the cube root of 250

Take 250, a number that is not a perfect cube, and work it through by hand.

1. Bracket it. The perfect cubes around 250 are 216 (6 cubed) and 343 (7 cubed). So the answer is between 6 and 7, and much nearer 6, because 250 is only 34 above 216 but 93 below 343.
2. Factor out the largest perfect cube. 250 = 125 × 2, and 125 is 5 cubed. So the cube root of 250 = the cube root of 125 times the cube root of 2 = 5∛2. That is the exact answer.
3. Convert to a decimal. The cube root of 2 is 1.25992105, and 5 × 1.25992105 = 6.29960525.
4. Check it. 6.29960525 × 6.29960525 × 6.29960525 = 250.000000, confirming the root.

Both answers are correct and they serve different purposes. Homework and exams normally want the radical 5∛2, because it is exact. A workshop measurement wants 6.2996 inches, because you cannot cut a board to an irrational length.

Second worked example: a negative radicand

Now try −1,728. A square root would stop here, because no real number squared is negative. A cube root does not, because three negative factors multiply to a negative product. Since 12 × 12 × 12 = 1,728, it follows that −12 × −12 × −12 = −1,728, so the cube root of −1,728 is exactly −12. The calculator flags it as a perfect cube and returns the whole number with no decimal at all.

The same logic works when the perfect cube is only part of the number. The cube root of −250 is −5∛2, or −6.29960525: the sign travels straight through the radical for any odd root. Try the same input with the index set to 2 or 4 and the calculator will tell you there is no real result, which is the mathematically honest answer rather than an error.

Perfect cubes from 1 to 30

Memorizing the first handful of cubes turns most cube-root questions into instant recall. This table lists every cube from 1 to 30 alongside its root, all computed directly.

Read the table backwards and it becomes a cube-root table: the cube root of 4,096 is 16, the cube root of 12,167 is 23, and the cube root of 27,000 is 30. Anything that is not in the right-hand columns is not a perfect cube, so its root is irrational.

Cube roots of 1 through 20

The other direction matters just as often: you have a small whole number and need its cube root as a decimal. Only 1 and 8 come out exactly here; every other row is irrational and rounded to six decimals.

Two entries are worth remembering. The cube root of 2 is 1.259921, the factor that doubles a volume, and the cube root of 10 is 2.154435, which lets you scale any cube root by powers of ten: the cube root of 1,000 is exactly 10, the cube root of 1,000,000 is exactly 100, and the cube root of 100 is 4.641589.

Simplified radical form, worked out

Simplifying a cube root means writing it as a whole number times a smaller radical. The rule is the product rule for radicals: the cube root of (a × b) equals the cube root of a times the cube root of b. Find the largest perfect-cube factor, take its root, and leave the rest inside. These are the cases that come up most often in algebra homework.

Notice the last row. Because 2,000 is 1,000 times 2, its cube root is exactly ten times the cube root of 2. Any factor of 1,000 shifts the decimal point of a cube root by exactly one place, which is a handy sanity check when you scale between cubic inches, cubic feet and cubic yards.

How to use this cube root calculator

1. Pick the mode. Cube root is selected by default, which fixes the index at 3. Choose the nth root card if you need a fourth root, a fifth root or anything up to a twentieth root.
2. Set the index if you are in nth root mode. Type a whole number from 2 to 20, or tap one of the shortcut chips.
3. Enter the number. Whole numbers, decimals and negatives are all accepted. Commas are ignored, so you can paste 1,728 straight in.
4. Read the headline figure. The large number is the real root. The badge under it says whether the value is exact or a rounded approximation of an irrational number.
5. Check the radical. The tile beside the result gives the exact simplified form, and the panel below shows the factoring that produced it.
6. Verify. The bottom card raises the answer back to the power you chose, so you can see that it reproduces your input.

Everything updates as you type, and nothing leaves your browser, so you can run through a whole homework set without reloading the page.

Who this calculator is for

• Algebra and pre-calculus students who need both the decimal and the simplified radical form for the same problem.
• Teachers and tutors building worked examples, since the calculator shows the perfect-cube factoring rather than just an answer.
• Makers, woodworkers and packaging designers converting a target volume into the edge length of a cube-shaped box or bin.
• Engineers and lab technicians going from a tank capacity back to a radius, or applying a cube-law scaling rule.
• Anyone comparing three-period growth, where the cube root of the total growth factor gives the average rate per period.

Where cube roots show up in practice

The single most common use is turning a volume back into a length. A cube with a volume of 500 cubic inches has an edge of 7.937005 inches, because that is the cube root of 500. A shipping crate holding 64 cubic feet is 4 feet on a side. A container of 2 cubic yards would be 1.2599 yards on a side, which is the cube root of 2 again in different clothing.

Spheres work the same way through their volume formula, V = (4/3)πr³. Rearranged, the radius is the cube root of 3V divided by 4π. A spherical tank holding 904.78 cubic units has a radius of almost exactly 6.0 units, and one holding 100 cubic units has a radius of 2.879412 units. Any time a quantity depends on the cube of a dimension, recovering that dimension means taking a cube root.

Growth rates over three periods are the finance version. If revenue rises from $1,200,000 to $1,800,000 over three years, the total growth factor is 1.5, and the average annual factor is the cube root of 1.5 = 1.144714, an average of 14.47% per year. That is a compound annual growth rate with n = 3, and it is the same arithmetic the calculator performs.

Cube roots of decimals and fractions

Decimals behave exactly like whole numbers, and some of them come out clean. The cube root of 0.125 is 0.5, because 0.5 cubed is 0.125. The cube root of 0.008 is 0.2, and the cube root of 0.001 is 0.1: each factor of 1,000 in the radicand moves the root by one decimal place. Less tidy inputs give irrational results, so the cube root of 2.5 is 1.35720881 and the cube root of 1.5 is 1.14471424.

For a fraction, take the cube root of the numerator and the denominator separately. The cube root of 8/27 is 2/3, because 2 cubed is 8 and 3 cubed is 27. If neither part is a perfect cube, convert to a decimal first and read the answer off the calculator. Radical simplification is defined for whole numbers, so the calculator gives the decimal only when the input has a fractional part.

Beyond the cube root: any nth root

Switch the mode and the same machinery handles any index. The fifth root of 32 is 2, because 2 to the fifth power is 32. The fourth root of 81 is 3, the fourth root of 256 is 4, and the sixth root of 729 is 3. The pattern is worth internalizing: odd indexes accept negative inputs and give exactly one real answer, while even indexes refuse negatives and give a positive principal root plus its negative twin.

Simplification generalizes as well. To simplify a fifth root you pull out perfect fifth powers instead of perfect cubes, so the fifth root of 64 becomes 2 times the fifth root of 2, since 64 = 32 × 2 and 32 is 2 to the fifth. The calculator applies whichever index you set, which is why the same tool covers both the cubed-root question and every higher root you are likely to meet.

Key terms

• Radicand: the number under the radical sign, the value whose root you are taking.
• Index: the small number in the notch of the radical. It is 3 for a cube root and 2 for a square root, where it is normally left off.
• Principal root: the root a calculator returns by default. For odd indexes it is the single real root; for even indexes it is the non-negative one.
• Perfect cube: an integer of the form k³, so its cube root is a whole number.
• Irrational number: a value that cannot be written as a fraction of whole numbers. Every cube root of a non-perfect-cube integer is irrational.
• Simplified radical form: the exact answer written as a whole number multiplied by the smallest possible radical, such as 5∛2.
• Fractional exponent: the equivalent power notation, x^1/3, which is how software actually computes a cube root.

What changes the result

• The index. Raising the index pulls every root toward 1: the cube root of 100 is 4.641589, the fourth root is smaller, and each further step shrinks it again.
• The sign of the radicand. With an odd index the sign passes straight through the radical. With an even index a negative input has no real answer at all.
• Whether the number is a perfect cube. Perfect cubes give exact integers; everything else gives an unending decimal that has to be rounded somewhere.
• Scale. Multiplying the radicand by 1,000 multiplies the cube root by exactly 10, which is why cubic-unit conversions are so easy to sanity check.
• Rounding of the input. Because the cube root grows slowly, a small change in the radicand barely moves the root, so a rounded volume still gives a usable length.

Tips for working with cube roots

• Memorize the cubes up to 10. Recognizing 216 and 343 instantly is what makes bracketing an estimate fast.
• Always look for a perfect-cube factor before reaching for a decimal. If the number is even, test 8; if it ends in 0 or 5, test 125; if the digits sum to a multiple of 9, test 27.
• Give the radical form on written work and keep the decimal for measurement. Rounding early and then multiplying compounds the error.
• Remember that cubing is the check. If the answer cubed does not return your input, something in the setup is wrong.
• To scale a shape without changing its proportions, apply the cube root of the volume ratio to every edge. Doubling a volume means multiplying each edge by 1.259921, an increase of about 25.99%.

Limitations and assumptions

• Only real roots are returned. In the complex numbers every non-zero value has three cube roots; this tool shows the single real one.
• Even roots of negative numbers are reported as having no real value rather than as an imaginary number.
• Decimal results are rounded for display. Irrational roots have infinitely many digits, so any decimal shown is an approximation.
• Arithmetic uses double precision, reliable to about 15 significant digits. Extremely large inputs lose accuracy in the final digits.
• Radical simplification runs on whole numbers within a safe factoring range; very large integers and decimals are answered as decimals only.

Which root calculator should you use?

Use this page for cube roots and higher nth roots, especially when you want the simplified radical form. For index 2 and the perfect-square checks that go with it, the Square Root Calculator is the direct sibling. Going the other way, from a base and an exponent to a power, is the job of the Exponent Calculator. If your radicand is enormous or tiny, the Scientific Notation Calculator makes it readable first, and the Scientific Calculator handles a longer expression with a root inside it. If your cube root came out of a container problem, the Volume Calculator works the same relationship in the forward direction.

Sources

Cube roots and nth roots are pure mathematics: the definition (y is the cube root of x exactly when y³ = x), the identity ∛x = x^1/3, and the product rule for radicals are exact statements that need no external citation, and every figure on this page was computed directly from them. The unit examples follow the exact US customary definitions maintained by the National Institute of Standards and Technology:

• National Institute of Standards and Technology (NIST) - Special Publication 811, Guide for the Use of the International System of Units (exact inch, foot and yard definitions used in the volume examples).
• National Institute of Standards and Technology (NIST) - Weights and Measures unit conversion reference.