Cube Root Calculator

∛x = ∛(m³ · r) = m · ∛r ∛(n ÷ d) = ∛(n · d²) ÷ d

x

The radicand, the number under the radical sign. It may be negative, because a cube root is an odd root and an odd power of a negative is still negative: the cube root of -27 is -3 and the cube root of -1 is -1. It may also be a decimal, up to three places, in which case the page turns it back into a fraction before simplifying it.

m³

The largest perfect cube that divides the radicand: 8, 27, 64, 125 and so on. Finding it is the whole of the simplification, and the page searches from the top down so the first factor it finds is the largest one. For 72 that factor is 8, which leaves 9 behind under the radical.

m

The cube root of that perfect cube, which comes out in front of the radical sign. When it is a whole number the exact form reads 2∛9; when it is 1 the coefficient is dropped and the form reads ∛2, because 1∛2 is not how anyone writes it.

r

Whatever is left under the radical once every cube factor has been pulled out. It is cube-free, which is exactly what makes the form final. When nothing is left — when the radicand was a perfect cube to begin with — the radical sign disappears entirely and the exact form is a plain integer, as with the cube root of 27.

d²

The square of the denominator, and the reason the decimal case is not simply copied from the square root page. To clear a denominator of d from under a cube root you need d cubed inside it, so n ÷ d is rewritten as (n · d²) ÷ d³ and the cube root of that is ∛(n · d²) ÷ d. Using d instead of d² still produces a number, just the wrong one.

Use this page when you need the exact form of a cube root, or when the radicand is negative. The cube calculator covers the forward direction — cubing a number and reading the root back — and it prints a decimal where this page prints a notation. When the index is not three, the general root calculator takes any index from 2 to 12 and simplifies it the same way.

1. Simplifying the cube root of 72

   1. Look for the largest perfect cube that divides 72
   2. 8 divides it: 72 ÷ 8 = 9, so 72 = 8 × 9
   3. The cube root of 8 is 2, which comes out in front: ∛72 = 2∛9
   4. Evaluating that exact form gives 4.160168

   The check is one line: 2∛9 cubed is 2³ × 9, which is 8 × 9, which is 72. The decimal column cannot be checked that way — 4.160168 cubed comes back to 71.99999 and change — which is the whole argument for printing the exact form beside it.

2. A perfect cube collapses to an integer

   1. 27 is 3 × 3 × 3, so the largest cube factor is 27 itself
   2. Pulling it out leaves nothing under the radical: 27 = 27 × 1
   3. With nothing left underneath, the radical sign goes away and the exact form is 3
   4. The decimal is 3 as well, so both columns agree

   When the radical empties, that is the signal that the radicand was a perfect cube. It is worth knowing that this row is the one a computer gets wrong by default: raising 27 to the power of one third in floating point gives 3.0000000000000004, and the page corrects it by checking whether a nearby whole number cubes back to 27.

3. A negative radicand

   1. Take the sign out first: ∛-72 = -∛72
   2. The positive part simplifies exactly as before: ∛72 = 2∛9
   3. Putting the sign back gives -2∛9
   4. The decimal is -4.160168, the negative of the positive case

   The minus sign goes at the front rather than inside the radical. -2∛9 and 2∛-9 are the same number, but only the first is the standard way to write it, and printing the second would look like a different expression. This is the case the square root calculator cannot handle at all: an even root of a negative number is not a real number.

4. A decimal radicand that is a perfect cube

   1. 0.125 as a fraction is 1/8
   2. Both parts are perfect cubes: ∛1 = 1 and ∛8 = 2
   3. Nothing is left under the radical, so the exact form is the fraction 1/2
   4. As a decimal it is 0.5

   This row is the reason the page accepts three decimal places rather than two. One eighth is the most natural decimal a cube root ever meets, and at two places it could not even be read as a fraction, so a reader would have been turned away from a cube root page by the number that most deserves to be let in.

The radicand is limited to three decimal places. A fourth place would be refused rather than rounded, because the simplification works by turning the decimal into a fraction and then trial-dividing, and the arithmetic stops being reliable past that point — the integer products involved would exceed what a double can represent exactly, so the answer would be quietly wrong rather than merely slow. The magnitude is limited to 1000000 in either direction, up to which the trial division stays cheap. The exact form is a notation rather than a number and is written identically in every language, since a radical sign and a division sign carry no words; the decimal beside it is rounded to six places. Only the real cube root is given: every non-zero number has three cube roots in the complex plane and the other two are not shown. The cube root of a negative number is printed with the sign in front, which is the standard spelling but is not the same string as a radical over a negative radicand.

What does a cube root calculator do?

It takes a radicand and returns two things: the decimal value of its cube root, and the exact form with the largest perfect cube pulled out in front of the radical sign. The cube root of 72 is 4.160168 as a decimal and 2∛9 in exact form, and the two are the same number written two ways.

How do I simplify a cube root by hand?

Look for the largest perfect cube that divides the radicand, pull it out, and take its cube root in front of the sign. For 72 that factor is 8, and since 72 ÷ 8 = 9 the whole thing becomes 2∛9. You know you are finished when nothing cube-free is left under the radical — for 27 the radical empties completely and the answer is just 3.

Can the radicand be negative?

Yes, and that is the clearest difference between this page and the square root calculator, which refuses negatives outright. A cube root is an odd root, so a negative radicand is fine: the cube root of -64 is -4. The sign is factored out first and printed at the front of the exact form, so -72 simplifies to -2∛9 rather than to a radical over a negative number.

Why does the exact form sometimes come out as a fraction?

When the radicand is the cube of a fraction, nothing is left under the radical and the exact form collapses to a plain fraction — the cube root of 0.125 is 1/2, and the cube root of 0.001 is 1/10. The radical survives only when something cube-free is left underneath it, the same rule that makes the cube root of 27 a plain integer.

How many decimal places does the radicand take?

Three, which is one more than the square root page allows. The extra place exists for a specific number: 0.125 is one eighth, the cube root of 0.5, and at two decimal places it could not be recognised as a fraction at all. A fourth decimal is refused rather than rounded, because past that point the trial division stops being exact.

• Cube Root — the inverse of cubing, defined on negative inputs because the index is odd — Wolfram MathWorld (United States)
• Cubic Number — the perfect cubes the simplification searches for, from 8 up to whatever the radicand allows — Wolfram MathWorld (United States)
• Radical — the radical sign, the radicand, and the convention for writing a coefficient in front of a root — Wolfram MathWorld (United States)