Cube root formula (and why it works)

The cube root is the inverse of “cubing.” If a number y is cubed, you multiply it by itself three times: y³ = y × y × y. The cube root reverses that operation. So when we say y = ∛x, we mean:

• y³ = x
• y = x^(1/3) (another common way to write cube root)

That exponent form matters because it connects cube roots to exponent rules you already use for square roots and powers. In general, x^(1/n) means the “nth root” of x. For cube roots, n = 3. This also explains a handy identity:

• Undoing a cube: ∛(y³) = y
• Redoing a root: (∛x)³ = x

Cube roots show up any time you’re working with volume and want to recover a length. A classic example is a cube: if a cube has volume V, then the side length is ∛V. So if the volume is 125 cubic units, the side length is ∛125 = 5 units.

• Example 1: ∛27
  Look for a number that cubed equals 27. Since 3³ = 27, the cube root is 3.
• Example 2: ∛(−8)
  Cube roots keep the sign. Since (−2)³ = −8, the cube root is −2.
• Example 3: ∛0.125
  Rewrite as a fraction: 0.125 = 125/1000 = 1/8. Then ∛(1/8) = ∛1 / ∛8 = 1/2 = 0.5.
• Example 4: ∛50 (not a perfect cube)
  50 sits between 3³=27 and 4³=64, so ∛50 is between 3 and 4. Numerically it’s about 3.684... (your calculator will show the rounded value you choose).

In your browser, we compute the real cube root using a sign-safe method: take the absolute value, compute abs(x)^(1/3), and then re-apply the sign. That’s why negative inputs produce negative outputs cleanly. After that, we optionally round to your chosen decimals and check whether the original number is close to a perfect cube (within a tiny tolerance to handle floating-point rounding).

If you’re using this for homework, two quick habits will save you time: (1) always sanity-check the cube by cubing your answer; and (2) compare your input against nearby perfect cubes. If you input 216 and you see a result near 6, that’s correct because 6³ = 216.

How cube roots behave (the parts people trip on)

Cube roots are friendlier than square roots in one key way: every real number has a real cube root. That means you can safely input a negative number and still get a real output. (Square roots of negative numbers require complex numbers, which is a different topic.)

If x is negative, then ∛x is also negative. Example: ∛(−64) = −4 because (−4)³ = −64. This is why the calculator never throws an error for negative inputs.

Cube roots grow much more slowly than the original number. If you multiply a number by 8, the cube root doubles: ∛(8x) = 2∛x. This is a neat “volume vs length” relationship: 8× volume corresponds to 2× side length in a cube.

If a number is a perfect cube (like 27, 64, 125), the cube root is an integer. If it isn’t (like 50 or 2), the cube root is usually an irrational decimal that never truly ends. So rounding is normal. The key is to round to the precision your problem expects.

Many decimals are “nice” cube roots because they’re fractions in disguise. Example: 0.001 = 1/1000, and ∛(1/1000) = 1/10 = 0.1. If your input has 3, 6, 9… zeros after the decimal, it might be a perfect cube fraction.

When simplifying radicals, you often factor out perfect cubes: ∛(54) = ∛(27×2) = ∛27 · ∛2 = 3∛2. That’s the cube-root version of pulling perfect squares out of a square root.

Bottom line: use this tool for quick answers, but use these rules to understand why your answer makes sense. That’s how you avoid “looks right” math mistakes.

Cube Root Calculator FAQs

• What is the cube root of a number?

  The cube root of x is the number y such that y³ = x. It’s written as ∛x. For example, ∛27 = 3 because 3³ = 27.

• Can the cube root of a negative number be real?

  Yes. Every real number has a real cube root. For example, ∛(−8) = −2. This is different from square roots, where negative inputs lead to complex numbers.

• Why do I get a long decimal?

  If the number isn’t a perfect cube, its cube root is often irrational (a non-terminating, non-repeating decimal). That’s why rounding is normal. Choose decimals based on your context (homework, measurement precision, engineering tolerance).

• How do I know if my number is a perfect cube?

  A quick check is to estimate the answer (between two known cubes) and see if the result is extremely close to an integer. This calculator also labels perfect cubes when the input is within a tiny tolerance of an integer cube.

• Is ∛x the same as x^(1/3)?

  For real numbers, yes: ∛x = x^(1/3). In calculators and code, the exponent form is common. This page uses a sign-safe approach so negative values still behave correctly.

• How can I verify the result?

  Cube the answer. If your result is y, compute y³. If you rounded, you’ll get something very close to your original x. Increasing decimals will make the check even closer.

• What’s the fastest mental cube root trick?

  Memorize small cubes (1–12). Then bracket the input between nearby cubes. Example: 400 is between 7³ = 343 and 8³ = 512, so ∛400 is between 7 and 8.