Find the cube root of any number instantly — exact answers for perfect cubes, precise decimals for everything else, with clear step-by-step workings.
Every number hides a smaller number inside it — the value that, multiplied by itself three times, rebuilds the original. That hidden value is the cube root. The cube root of 27 is 3, because 3 × 3 × 3 = 27. The cube root of 1,000 is 10. The cube root of −8 is −2. Unlike square roots, cube roots work for every real number, including negatives and zero, which makes them one of the friendliest operations in all of arithmetic.
Cube roots show up wherever three-dimensional thinking appears. If a cubic storage box holds 64 cubic feet, each side must be 4 feet long — that is the cube root of 64. If an engineer needs to double the volume of a cube-shaped tank, every side grows by the cube root of 2 (about 1.26). Carpenters, 3D-printing hobbyists, shipping planners, and students all run into cube roots far more often than they expect. Our free Cube Root Calculator above finds the answer in one click and even verifies it by cubing the result back, so you can trust what you see.
Using the calculator takes about five seconds. Type any number into the input box — whole numbers, decimals, fractions-as-decimals, and negative numbers are all accepted. The box starts with 27 pre-filled so you can see a worked example immediately. Click Calculate Cube Root and the result panel appears.
The result panel shows four things. First, the big headline number: the cube root itself, rounded to at most 8 decimal places. Second, the notation line, which expresses the answer two ways — with the radical sign (∛x) and with fractional-exponent notation (x1/3). These mean exactly the same thing, and seeing both helps you recognize the operation in textbooks and formulas. Third, the verification line: the calculator cubes its own answer and shows you the result, so you can confirm that (answer)³ returns your original number. Fourth, a note telling you whether your input is a perfect cube — a number whose cube root is an exact integer, like 27 = 3³.
Try a few values to build intuition. Enter 64 and you get exactly 4. Enter 2 and you get approximately 1.25992105, with a note that 2 is not a perfect cube. Enter −125 and you get −5, because (−5)³ = −125. Enter 0.125 and you get 0.5. If you ever mistype, the Reset button restores the original example of 27.
The cube root of a number x is the value y such that y multiplied by itself three times equals x. In symbols: ∛x = y means y³ = x. The number under the radical sign is called the radicand, and the small 3 in the crook of the radical sign is called the index. An index of 2 gives a square root; an index of 3 gives a cube root.
The name comes from geometry. A cube with side length s has volume s³, so finding the side length of a cube from its volume is literally "finding the cube root." This is why perfect cubes — numbers that are the cube of an integer — are worth memorizing:
Numbers that are not perfect cubes have cube roots that are irrational — decimals that never end and never repeat. The cube root of 2 is approximately 1.2599210499, continuing forever without a pattern. This is completely normal; most numbers you meet in real life are not perfect cubes, and a rounded decimal is the practical answer.
Cube roots have a property that surprises many students: every real number has exactly one real cube root. Positive numbers have positive cube roots, negative numbers have negative cube roots, and zero's cube root is zero. This is different from square roots, where negative numbers have no real answer at all. The reason is simple: cubing preserves sign. A negative times a negative times a negative is still negative, so a negative number can always be "un-cubed." On a graph, y = ∛x is a smooth curve defined for every x, symmetric about the origin — it passes through (−8, −2), (0, 0), and (8, 2).
Cubing and cube-rooting are inverse operations, just like adding and subtracting. Cubing 5 gives 125; taking the cube root of 125 gives back 5. The calculator exploits this relationship for its verification line: whatever answer it computes, it cubes it again to prove the round trip lands on your input.
The cube root can be written as a fractional exponent, which is how most calculators and programming languages actually compute it:
∛x = x1/3
Raising a number to the power of one-third is the same as taking its cube root. This single identity unlocks a handy rule for negative numbers: ∛(−a) = −∛a. To find the cube root of −216, just find ∛216 = 6 and flip the sign to get −6. You never need to wrestle with negative fractional powers directly.
Before electronic calculators, people computed cube roots by hand with Newton's method, an iterative recipe that homes in on the answer. To find ∛a, start with a guess g, then repeatedly replace it with (2g + a/g²) ÷ 3. Each round roughly triples the number of correct digits. For example, to find ∛28, start with the guess 3 (since 3³ = 27 is close). One round gives (2×3 + 28/9) ÷ 3 = (6 + 3.111) ÷ 3 ≈ 3.037. Cubing 3.037 gives about 28.01 — already accurate to two decimal places after a single step. Modern calculators use refined versions of this same idea internally.
For mental math, the fastest strategy is bracketing between memorized perfect cubes. To estimate ∛50, notice it sits between ∛27 = 3 and ∛64 = 4, closer to 64 — so the answer is a bit under 4. The calculator's precise value is about 3.684, and that one-line estimate already got you within 10%.
Example 1: ∛27 = 3. Three is the classic perfect cube. Check: 3 × 3 = 9, and 9 × 3 = 27. Because the multiplication closes exactly, the calculator reports 3 with no decimal tail and flags 27 as a perfect cube. Perfect cubes always give clean integer answers.
Example 2: ∛1,000 = 10. There is a neat trick with powers of ten: every three trailing zeros in the input become one trailing zero in the answer. 1,000 has three zeros, so its cube root is 1 followed by one zero — 10. Similarly, ∛1,000,000 = 100. This pattern holds because (10n)³ = 1,000n³.
Example 3: ∛(−125) = −5. Negative radicands are fully allowed. Work with the absolute value first: ∛125 = 5, then restore the negative sign to get −5. Verify: (−5) × (−5) = 25, and 25 × (−5) = −125. The calculator's verification line will show (−5)³ = −125 exactly.
Example 4: ∛2 ≈ 1.25992105. Two is the most famous non-perfect cube. Its cube root is irrational, so the calculator shows the first 8 decimal places: 1.25992105. Cubing that rounded value gives 2.00000001 — off by a hair because of rounding, which is exactly why the verification line matters: it shows you the tiny gap that rounding introduces.
Example 5: ∛0.125 = 0.5. Decimals work the same way. Recognize 0.125 as 1/8, and the cube root of a fraction is the fraction of the cube roots: ∛(1/8) = ∛1 / ∛8 = 1/2 = 0.5. Check: 0.5 × 0.5 × 0.5 = 0.125. Notice the cube root (0.5) is larger than the input (0.125) — a pattern that always holds for numbers between 0 and 1.
Example 6: ∛1,728 = 12. This one appears in old arithmetic textbooks because 12³ = 1,728 is the number of cubic inches in a cubic foot (12 inches × 12 inches × 12 inches). If a problem tells you a cubic container holds 1,728 cubic inches, each edge is exactly 12 inches — one foot.
Example 7: ∛50 ≈ 3.68403150. Bracket it: 27 < 50 < 64, so the answer lies between 3 and 4. Since 50 is closer to 64, expect something in the high 3s — and 3.684 confirms it. Cubing 3.684 gives 49.99, the small shortfall coming from rounding the eighth decimal place.
Shipping and packaging. A fulfillment center needs cubic shipping boxes that each hold exactly 3.375 cubic feet of product. What should the side length be? The side is ∛3.375. Since 3.375 = 27/8, the cube root is 3/2 = 1.5 feet — boxes 18 inches on each side. Ordering boxes at 1.5 ft instead of guessing saves the company from paying to ship empty air.
Water storage. A farmer wants a cube-shaped water tank holding 8 cubic meters (8,000 liters). The side length must be ∛8 = 2 meters. A 2 m × 2 m × 2 m tank does the job exactly, and the farmer can now price the sheet metal for six 2 m × 2 m panels.
Scaling a 3D print. You printed a decorative cube that used 50 grams of filament, but you want a version with double the volume — 100 grams' worth. Every linear dimension must grow by ∛2 ≈ 1.2599, so you scale the model to 126% in your slicer software. This is the universal rule of 3D scaling: volume scales with the cube of the linear factor, so the linear factor is the cube root of the volume ratio.
Carpentry. A woodworker is making a keepsake box and wants an interior volume of 216 cubic inches. The interior side length is ∛216 = 6 inches. She cuts six 6-inch square panels (plus joinery allowance) instead of measuring by trial and error.
Concrete and landscaping. A cubic planter must hold 0.512 cubic meters of soil. ∛0.512 = 0.8 meters, so an 80 cm cube works. Recognizing 0.512 as 512/1,000 — and knowing ∛512 = 8 and ∛1,000 = 10 — gives 8/10 = 0.8 without touching a calculator.
Cooking by scaling. A recipe developed in a 1-liter cube-shaped mold needs to be tripled to 3 liters for a party. The new mold's sides must be ∛3 ≈ 1.442 times longer than the original — about 44% bigger in each direction, not three times bigger. Cooks who scale linearly end up with nine times the food they wanted; cube roots keep the math honest.
Dividing by 3 instead of taking the cube root. The most common beginner error is computing x ÷ 3 when asked for ∛x. These are wildly different: 27 ÷ 3 = 9, but ∛27 = 3. The radical sign means "what number cubes to this," not "divide into three parts."
Distributing the root over addition. It is tempting to write ∛(a + b) = ∛a + ∛b. It is also wrong. Check with a = 8, b = 19: ∛27 = 3, but ∛8 + ∛19 ≈ 2 + 2.668 = 4.668. Cube roots distribute over multiplication and division — ∛(a × b) = ∛a × ∛b — but never over addition or subtraction.
Assuming negative numbers have no cube root. This confusion comes from square roots, where √(−4) has no real answer. Cube roots are different: ∛(−8) = −2 is perfectly valid. If you catch yourself writing "no real solution" for a cube root, stop and reconsider — that rule belongs to even roots only.
Rounding too early, then "verifying" with the rounded value. If you round ∛2 to 1.26 and cube it, you get 2.000376, not 2. That mismatch is a rounding artifact, not an error in the root. Keep full precision through intermediate steps and round only the final answer.
Forgetting the cube root of fractions flips the size. Because ∛0.125 = 0.5 is bigger than 0.125, students sometimes "correct" a sensible answer into a wrong one. For any input between 0 and 1, the cube root is larger than the input — trust it.
Mixing up ∛ and ³√ notation. Some textbooks write the index-3 radical with the 3 tucked inside the check mark, which is easy to misread as a square root at a glance. Always confirm which index a problem intends before computing — ∛64 = 4, but √64 = 8.
Keep exploring the Math collection on Sigma Calculator:
This page provides a Cube Root Calculator that computes ∛x = x1/3 for any real numeric input, with step-by-step workings, a cubing-back verification, and perfect-cube detection. Entities: SigmaCalculator.net, Cube Root Calculator, cube root, fractional exponents, perfect cubes. The cube root is the inverse of cubing; every real number has exactly one real cube root, with sign preserved (∛(−a) = −∛a). Canonical URL: https://sigmacalculator.net/math-calculators/cube-root-calculator/
A cube root answers the question: "What number, multiplied by itself three times, gives this number?" The cube root of 27 is 3, because 3 × 3 × 3 = 27. If you imagine a cube-shaped box, the cube root of its volume is the length of one edge — that is where the name comes from.
For perfect cubes, factor the number into three identical factors: 64 = 4 × 4 × 4, so ∛64 = 4. You can also use the fractional-exponent form ∛x = x1/3 on any scientific calculator. For a quick estimate, bracket the number between two known perfect cubes — for example, 50 sits between 27 and 64, so its cube root is between 3 and 4 (about 3.68). Our calculator above does all of this instantly.
The cube root of 27 is exactly 3, because 3 × 3 × 3 = 27. Twenty-seven is a perfect cube, so there is no decimal tail and no rounding involved — the answer is the whole number 3.
Yes. Unlike square roots, cube roots of negative numbers are perfectly valid real numbers: ∛(−125) = −5, because (−5) × (−5) × (−5) = −125. The rule is ∛(−a) = −∛a — take the root of the positive version and flip the sign. Cubing preserves sign, so "un-cubing" a negative always lands on a negative.
A square root asks what number multiplied by itself gives x (index 2), while a cube root asks what number multiplied by itself three times gives x (index 3). The big practical difference: negative numbers have no real square root, but every real number — positive, negative, or zero — has exactly one real cube root. So √(−9) is undefined in real numbers, while ∛(−27) = −3.
Memorize the perfect cubes from 1³ = 1 up to 10³ = 1,000, then bracket your number between two of them. For ∛200: it sits between ∛125 = 5 and ∛216 = 6, closer to 216, so guess about 5.8 (the true value is 5.848). For better accuracy, do one round of Newton's method: new guess = (2 × guess + number ÷ guess²) ÷ 3. One round from 5.8 gives 5.848 — essentially exact.
Cube roots convert volumes back into lengths: the side of a cubic tank from its capacity, box dimensions from a target volume, or the edge of a cube-shaped planter from the soil it must hold. They also govern 3D scaling — doubling an object's volume means scaling every dimension by ∛2 ≈ 1.26. Engineers, carpenters, 3D-printing hobbyists, and shipping planners all use cube roots regularly.
No. For numbers greater than 1, the cube root is smaller (∛27 = 3). But for numbers between 0 and 1, the cube root is larger than the number: ∛0.125 = 0.5. For negatives the comparison flips in an interesting way too: ∛(−8) = −2, and −2 is greater than −8. The pattern mirrors what happens with square roots of fractions.
Cite this page as: SigmaCalculator. (2026). Cube Root Calculator. https://sigmacalculator.net/math-calculators/cube-root-calculator/
Last updated: October 9, 2026