Cube Root Calculator
A cube root calculator finds the number that, multiplied by itself twice, gives your number. The cube root of 27 is 3, because 3 × 3 × 3 = 27, and the cube root of 64 is 4, because 4 × 4 × 4 = 64. The square root calculator above handles cube roots and other roots in its root selector, and this page explains the cube root, shows how to find one by hand, and works through the perfect cubes worth knowing.
Cube roots differ from square roots in one useful way: they work for negative numbers too. The cube root of −8 is −2, because (−2) × (−2) × (−2) = −8, while the square root of a negative number has no real answer. Cube roots also appear wherever volume is involved: the side of a cube is the cube root of its volume, so a box holding 125 cubic centimetres has sides of 5 cm, because 5³ = 125.
The Square Root: What It Is and How to Find It
The square root of a number is the value that, multiplied by itself, gives the original number. The square root of 144 is 12, because 12 × 12 = 144, and the square root of 2 is about 1.41421, because 1.41421 × 1.41421 is very close to 2. The symbol is √, so √144 = 12 and √2 ≈ 1.41421. Every positive number has two square roots, one positive and one negative, but the symbol √ on its own always means the positive one, called the principal square root. The square root of a negative number is not a real number at all, which is why the calculator reports an error for negative inputs.
Most square roots are irrational numbers: their decimal expansion never ends and never repeats. The square root of 2 is the famous example — its digits run 1.41421356 and continue forever without a pattern, a fact proved by the ancient Greeks. That is why we round: √50 ≈ 7.07 is accurate to two decimal places, and √50 ≈ 7.071 is accurate to three. The calculator above also handles cube roots and other roots: the cube root of 27 is 3, because 3 × 3 × 3 = 27, and the fourth root of 81 is 3, because 3⁴ = 81.
Square roots appear wherever a square is involved: the side of a square from its area, the diagonal of a rectangle, the standard deviation in statistics, and the quadratic formula in algebra. The National Institute of Standards and Technology lists the square roots of small integers among its standard mathematical constants, and the same values turn up in geometry, physics, and engineering problems of every kind.
Step 1 — Check Whether the Number Is a Perfect Square
Try the squares you know: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. If your number is on that list, its square root is exact: √144 = 12, √225 = 15. If it is not, the root is irrational and you will estimate or use the calculator.
Step 2 — Estimate Between Two Whole Numbers
For √50, note that 49 is 7² and 64 is 8², so √50 sits just above 7. For √125, note that 121 is 11² and 144 is 12², so the answer sits just above 11, and indeed √125 ≈ 11.18. A good estimate tells you at a glance whether a calculator answer is plausible.
Step 3 — Refine and Check by Squaring
Refine the estimate by squaring your guess: 7.07² = 49.9849, just under 50, so √50 ≈ 7.07 is right to two decimal places. Always check by squaring the answer you report; if the square does not return your original number, something went wrong.
Worked Examples: Finding Cube Roots
Each row shows the number, the cube that produces it, and the cube root, with a check by cubing.
| Number | Cube | Cube root |
|---|---|---|
| 27 | 3 × 3 × 3 | 3 |
| 64 | 4 × 4 × 4 | 4 |
| 125 | 5 × 5 × 5 | 5 |
| 8 | 2 × 2 × 2 | 2 |
| 1,000 | 10 × 10 × 10 | 10 |
The table is the whole method for perfect cubes: recognise the cube, and the root follows. For numbers that are not perfect cubes, estimate between the two nearest cubes: the cube root of 54 sits between 3 and 4, because 3³ = 27 and 4³ = 64, and it is about 3.78, because 3.78³ = 54.01. The cube root of 2 is about 1.26, because 1.26³ = 2.00, and the cube root of 100 is about 4.64, because 4.64³ = 99.90.
How to Use the Square Root Calculator
Enter any number, choose the root, and the result appears instantly, with the exact simplified form where one exists. Use it for homework checks, geometry, and any calculation where a root is one step of a longer problem.
- Enter the number. Type the value whose root you want, such as 72. The default of 72 shows √72 ≈ 8.49, or 6√2 in exact form.
- Choose the root. Square root is the default; switch to cube root, fourth root, or any other root for problems such as ∛27 = 3.
- Read the decimal result. The calculator gives the root to several decimal places, which is the form you need for measurement and engineering work.
- Read the simplified form. Where the number has a square factor, the calculator shows the exact form: √72 = 6√2, √50 = 5√2, √125 = 5√5.
- Square the answer to check. Multiply the result by itself and confirm you get back to the number you entered; that one check catches almost every input error.
Perfect Cubes to Memorise
These cubes are worth knowing by heart. They make cube root problems instant, and they anchor every estimate of a non-perfect cube.
| Number | Cube root | Check |
|---|---|---|
| 1 | 1 | 1³ = 1 |
| 8 | 2 | 2³ = 8 |
| 27 | 3 | 3³ = 27 |
| 64 | 4 | 4³ = 64 |
| 125 | 5 | 5³ = 125 |
| 216 | 6 | 6³ = 216 |
| 343 | 7 | 7³ = 343 |
| 1,000 | 10 | 10³ = 1,000 |
Common Mistakes With Square Roots
Square roots are a classic source of small, stubborn errors. These are the ones to watch for.
- Adding roots as if they were numbers. √2 + √3 is not √5. Roots add only when the radicand matches: √2 + 3√2 = 4√2.
- Forgetting the negative root. The equation x² = 49 has two solutions, x = 7 and x = −7, even though √49 on its own means 7.
- Taking the root of a negative number. There is no real number whose square is negative, so √(−4) has no real answer and the calculator correctly refuses it.
- Rounding too early. Rounding √2 to 1.41 and then squaring gives 1.9881, not 2. Keep extra digits until the last step of the problem.
- Confusing square and square root. Squaring makes numbers bigger than 1 larger; the square root makes them smaller. √144 = 12, while 144² = 20,736.
Where You Will Use Square Roots
Square roots are not just a school topic. They appear in practical work wherever area, distance, or variation is measured.
- Geometry. The side of a square garden with an area of 144 square metres is √144 = 12 metres, and the diagonal of a 3 by 4 rectangle is √(9 + 16) = 5.
- Statistics. The standard deviation is the square root of the variance, so every report of spread in test scores or measurements runs through a square root.
- Physics. The speed of a falling object, the period of a pendulum, and the energy in a spring all involve square roots of the quantities you measure.
- Construction. The 3-4-5 triangle, checked with √(3² + 4²) = 5, is how builders square up corners on site.
- Finance. The square root of time rule scales risk and volatility: over four times the period, typical movement roughly doubles, because √4 = 2.
Cube Roots and Volume
The cube root is the volume operation in reverse: volume is side cubed, so side is the cube root of volume. A cube with a volume of 64 cubic centimetres has sides of 4 cm, a cube with a volume of 216 cubic centimetres has sides of 6 cm, and a cube with a volume of 1,000 cubic centimetres has sides of 10 cm. For a volume that is not a perfect cube, such as 100 cubic centimetres, the side is about 4.64 cm, because 4.64³ = 99.90. The same idea scales to any shape whose volume grows with the cube of its size: doubling the side multiplies the volume by eight, because 2³ = 8.
Tips for Accurate Results
A little care with roots pays off, because a root error hides inside every later step of a problem.
- Memorise the squares up to 15². Knowing 12² = 144 and 15² = 225 lets you spot perfect squares instantly and estimate everything else.
- Simplify before approximating. Reducing √72 to 6√2 keeps the answer exact, and the decimal 8.49 follows from it without extra rounding error.
- Keep two extra digits. Carry two more decimal places than your final answer needs, and round only at the end.
- Check with a square. Squaring your answer is the fastest possible check, and it takes two seconds.
- Watch the domain. The input must be zero or positive for a real square root; a negative input means the problem needs a different approach.
Square Root Quick Answers
These are the root values people look up most often, each one verified against the calculator above.
| Question | Answer |
|---|---|
| Square root of 144 | 12 |
| Square root of 50 | 7.07 |
| Square root of 2 | 1.41421 |
| Cube root of 27 | 3 |
| Cube root of 64 | 4 |
Related tools on this site: the main square root calculator, scientific calculator, basic calculator, percentage calculator, and the square root calculator again whenever you need a fresh check.
Frequently Asked Questions
What is the cube root of 27?
The cube root of 27 is 3, because 3 × 3 × 3 = 27. It is a perfect cube, so the answer is exact, with no rounding and no decimal places.
How do I find a cube root by hand?
Check whether the number is a perfect cube first: 125 is 5³, so its cube root is 5. If it is not, estimate between the two nearest cubes and refine by cubing your guess. For 54, the neighbours are 27 and 64, so the root is between 3 and 4, and 3.78³ = 54.01 confirms it.
Can a cube root be negative?
Yes. The cube root of −8 is −2, because (−2) × (−2) × (−2) = −8. Cube roots work for negative numbers, unlike square roots, because a negative number cubed stays negative.
What is the cube root of 2?
The cube root of 2 is about 1.26, because 1.26³ = 2.00. It is irrational, so its decimal expansion never ends, and 1.26 is rounded to two decimal places.
How is a cube root different from a square root?
A square root asks which number squared gives your number, and it has no real answer for negatives. A cube root asks which number cubed gives your number, and it works for negatives too. The cube root of 64 is 4, while the square root of 64 is 8.
Disclaimer: This calculator is for general information only. Results are rounded for display; for engineering, construction, or safety-critical work, confirm measurements with the applicable standards and a qualified professional.