What does a cube calculator do that a plain multiplication does not?

It gives both directions at once, and it keeps them exact. Typing 27 returns the cube 19683 and, beside it, the cube root 3 — the value that would have to be cubed to get 27 back. Doing that by hand is two separate calculations and one of them, the root, is the one people get wrong.

Which numbers are perfect cubes?

1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331 and 1728 — the cubes of the whole numbers from 1 to 12, which is exactly what the reference table below shows. The test is not memorisation: type the number in and look at the root. A whole number there means a perfect cube, and a decimal means it is not.

Can I cube a negative number, and can I take its cube root?

Both, and the answer is negative both times. Multiplying three negatives gives a negative, because an odd count of minus signs survives, so the cube of -4 is -64 and the cube root of -64 is -4. An even root could not do this — the square root of a negative number is not a real number — which is why the square root calculator refuses negatives and this page does not.

Why does a large result switch to scientific notation?

Because nineteen digits of zeros tell you nothing. The cube of 1000000 is 1 followed by eighteen zeros, and printed out in full it wraps across the panel without making the magnitude any clearer. Written as 1.000000 × 10¹⁸ the size is immediate. Small results are left alone: 343 is printed as 343.000000, not as a power of ten.

How is this different from the cube volume calculator?

One takes a number, the other takes a length. Here the input is a pure value with no units, so 5 cubed is 125 and that is the end of it. The cube volume calculator reads the same 5 as a side length in centimetres and replies with 125 cm³ of volume and 150 cm² of surface area — the same arithmetic with a different question attached.

Cube Calculator

Result

343.000000

Cube

Cube root: 1.912931
Expanded form: 7 × 7 × 7

A cube calculator raises a number to the third power and shows the multiplication it stands for, so 7 cubed comes back as 343 beside 7 × 7 × 7. The same screen answers the reverse question, printing the cube root of that number: 3 when you type 27, 7 when you type 343. A whole-number cube root is the signal that the value you entered is a perfect cube, and a decimal there is equally informative. Negative numbers are accepted on both sides, because cubing a negative gives a negative and the cube root of a negative number is negative as well, so the minus sign survives instead of being refused. Results that would run off the panel are rewritten in scientific notation rather than padded out with zeros.

The cubes of the whole numbers from 1 to 12

Value	Cube
1	1
2	8
3	27
4	64
5	125
6	216
7	343
8	512
9	729
10	1000
11	1331
12	1728

Every row here is exact, and the last one is the reason the page prints large results in scientific notation: 12 cubed is 1728 and still comfortable, but the same form carried up to a million would not be. The rows that most often get looked up are 2, 3, 4 and 5 — 8, 27, 64 and 125 — because those are the perfect cubes that turn up in factorising and in geometry problems. Set beside the reference table on the cube root page, this one is the forward direction and that one is the reverse, which is the pair of questions the page answers.

Formula

x³ = x × x × x ∛x = x^(1/3)

x: The number being cubed. It may be negative and it may carry decimals, up to six of them, anywhere from -1000000 to 1000000. There is no positivity guard on this page because cubing a negative number is perfectly well defined: -4 times -4 times -4 is -64. The same is true of the cube root underneath, which is an odd root and therefore defined for negative inputs as well.
x³: The cube itself, printed to six decimals. Above the display window it is written in scientific notation so the magnitude stays readable: the cube of 1000000 is 1.000000 × 10¹⁸ rather than a nineteen-digit integer running across the panel. Inside the window the ordinary form is kept, so 343 is printed as 343.000000 and not as a power of ten.
∛x: The cube root of the same number, which undoes the first column: it is the value that gives x back when multiplied by itself twice. When x is a perfect cube the root is an exact whole number and no rounding is involved; otherwise it is rounded to six decimals. This is the difference between an odd root and an even one, and the reason a negative input is fine here while the square root of a negative number has no real answer at all.

Reach for this page when the question is about a number rather than a shape: what is 7 cubed, what is the cube of 1.5, what is the cube root of 343. When the cube stands for a physical object — a box with a side length and a volume in cubic centimetres — the cube volume calculator takes the edge length and answers in units instead.

Worked examples

Seven cubed, and the root that comes back
1. Square it first: 7 × 7 = 49
2. Multiply by 7 once more: 49 × 7 = 343
3. The cube root of 343 is 7, because 7 × 7 × 7 = 343
4. The cube root of the input 7 itself is 1.912931, which is where the six decimals come from — 7 is not a perfect cube
The two halves of the page answer different questions and both are worth having. The cube 343 is exact, and the root that leads back from it is exact too. The root of the input, 1.912931, is the rounded one: multiplying it by itself three times returns 6.999999 and change, which is as close as six decimals can get to 7.
A perfect cube: 27
1. 27 × 27 = 729
2. 729 × 27 = 19683
3. The cube root of 27 comes back as exactly 3, with nothing after the decimal point
4. That exact 3 is the whole answer to the question of whether 27 is a perfect cube
A perfect cube is the one case where the root column carries no rounding at all, and it is worth seeing why that matters numerically: a computer working out 27 to the power of one third gets 3.0000000000000004, which would print as 3.000000 and look like a rounding failure. The page tidies that away by checking whether a neighbouring whole number cubes back to the input.
Cubing a negative number
1. -4 × -4 = 16, a positive number, because two negatives multiply to a positive
2. 16 × -4 = -64, negative again, because the third factor is still negative
3. So the cube of -4 is -64, and the cube root of -64 is exactly -4
4. The cube root of the input -4 is -1.587401
Three factors is what decides the sign: an odd number of negatives gives a negative product and an even number gives a positive one. That is also why the cube root takes negatives while the square root does not — an odd root can be undone, an even root cannot. The minus sign is printed at the front of the root, the way it is written by hand.

Limitations

The input is limited to 1000000 in either direction. That is a display limit rather than a mathematical one: the cube of 1000000 is 1000000000000000000, which is still an exactly representable double, so nothing overflows at the boundary — the page simply refuses to print a number that a reader cannot take in at a glance. Below the limit the cube is exact for whole numbers and correct to six decimals otherwise. The cube root is a single real value, not the set of three cube roots a complex-number treatment would give: the real root of a negative input is the negative one, and the two complex roots are not shown. The expansion column is a notation for the multiplication and is written the same way in every language. If what you want is the volume of a box rather than the cube of a number, the cube volume calculator is the page that takes an edge length with units.

Frequently asked questions

What does a cube calculator do that a plain multiplication does not?: It gives both directions at once, and it keeps them exact. Typing 27 returns the cube 19683 and, beside it, the cube root 3 — the value that would have to be cubed to get 27 back. Doing that by hand is two separate calculations and one of them, the root, is the one people get wrong.
Which numbers are perfect cubes?: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331 and 1728 — the cubes of the whole numbers from 1 to 12, which is exactly what the reference table below shows. The test is not memorisation: type the number in and look at the root. A whole number there means a perfect cube, and a decimal means it is not.
Can I cube a negative number, and can I take its cube root?: Both, and the answer is negative both times. Multiplying three negatives gives a negative, because an odd count of minus signs survives, so the cube of -4 is -64 and the cube root of -64 is -4. An even root could not do this — the square root of a negative number is not a real number — which is why the square root calculator refuses negatives and this page does not.
Why does a large result switch to scientific notation?: Because nineteen digits of zeros tell you nothing. The cube of 1000000 is 1 followed by eighteen zeros, and printed out in full it wraps across the panel without making the magnitude any clearer. Written as 1.000000 × 10¹⁸ the size is immediate. Small results are left alone: 343 is printed as 343.000000, not as a power of ten.
How is this different from the cube volume calculator?: One takes a number, the other takes a length. Here the input is a pure value with no units, so 5 cubed is 125 and that is the end of it. The cube volume calculator reads the same 5 as a side length in centimetres and replies with 125 cm³ of volume and 150 cm² of surface area — the same arithmetic with a different question attached.

References

Cube — the third power of a number, the expansion x · x · x, and why the cube of a negative stays negative — Wolfram MathWorld (United States)
Cubic Number — the perfect cubes 1, 8, 27, 64 and the test for recognising one — Wolfram MathWorld (United States)
Cube Root — the inverse of cubing, defined for negative inputs because the index is odd — Wolfram MathWorld (United States)

Related calculators

Cube Root CalculatorCube root of a number, including negative inputs.Exponent CalculatorRaise a number to any power, including negative exponents.Cube Volume CalculatorVolume and surface area of a cube from its side.