Cube Root Calculator
Calculate the cube root of any real number, positive or negative, compute any nth root by changing the degree, or reverse-solve by entering a known root to find the original number. Each result shows a verification step and a plain-English explanation.
Formula
Worked example
For x = 27, n = 3: root = 27^(1/3) = 3. Check: 3^3 = 27. For x = -64, n = 3: root = -(64^(1/3)) = -4. Check: (-4)^3 = -64. Reverse-solve: enter y = 3, n = 3 to get x = 3^3 = 27.
What the cube root is and how to compute it
The cube root of a number x is the value y satisfying y^3 = x. It is written as the radical symbol with a small 3 above it, or equivalently as x raised to the power 1/3. For example, the cube root of 8 is 2 (since 2 x 2 x 2 = 8), and the cube root of 1,000 is 10. Geometrically, if a cube has a volume of x cubic units then each edge is the cube root of x units long. Calculators and programming languages typically use the cbrt() function or the exponent notation x^(1/3) to evaluate it to full floating-point precision.
Why negative numbers have a real cube root
A square root of a negative number is not real, because squaring any real number always yields a non-negative result. Cube roots behave differently: multiplying three negatives together keeps the product negative, so (-4) x (-4) x (-4) = -64. This means every real number, positive or negative, has exactly one real cube root, and that root shares its sign with the input. Even-degree roots (square, fourth root, sixth root...) require a non-negative input for a real result; odd-degree roots (cube, fifth, seventh...) work for any real number.
nth root mode: cube roots, fourth roots, and beyond
Changing the degree input lets you compute any integer root. The second root (n = 2) is the square root, the third root (n = 3) is the standard cube root, and higher degrees are used in finance (compound-growth calculations), physics (wave equations), and geometry (hypercube edge lengths). The general formula is the nth root of x equals x^(1/n). For even n the input must be non-negative for a real result; this calculator flags that condition rather than returning a complex value.
Reverse-solve: find a number from its known root
Sometimes you know the root but need the original number. Switch to reverse-solve mode, enter the root value y and the degree n, and the calculator returns x = y^n. For example, if you know the cube root is 5, the original number is 5^3 = 125. This is useful in chemistry (molar concentrations from dilution factors), finance (finding a principal from an interest factor), and geometry (reconstructing a cube volume from edge length).
Perfect cubes and estimating roots by hand
A perfect cube is an integer whose cube root is also an integer. The first ten are 1, 8, 27, 64, 125, 216, 343, 512, 729, and 1,000. Knowing these lets you bracket any cube root: because 50 lies between 27 (3^3) and 64 (4^3), its cube root must be between 3 and 4, closer to 4. A sharper estimate uses the last digit of the number: a perfect cube ending in 3 has a root ending in 7 (since 7^3 = 343), and a cube ending in 8 has a root ending in 2 (since 2^3 = 8). The reference table below lists these last-digit pairs for all single digits.
Real-world applications of the cube root
Cube roots appear in many fields. In geometry, the edge of a cube equals the cube root of its volume. In physics, the radius of a sphere of volume V is the cube root of (3V divided by 4 pi). In finance, the cube root of (final value divided by initial value) over three years gives the annual growth factor. In music, equal temperament divides an octave into 12 semitones, each a factor of the 12th root of 2. In chemistry, the van der Waals radius is proportional to the cube root of molecular volume.
Mental math: last-digit shortcut for perfect cubes
Every digit 0-9 maps to a unique last digit when cubed, unlike squares where 2 and 8 both end in 4. This means you can read the last digit of a perfect cube and immediately know the last digit of its root. For example, 117,649 ends in 9, so its cube root ends in 9 (it is 49). The table below shows all last-digit pairs. Combine that with the bracketing method above to narrow down cube roots of large perfect cubes quickly.
Perfect cube last-digit pairs and common cube roots
| Root digit (y) | Cube ends in | Example: y^3 |
|---|---|---|
| 0 | 0 | 0^3 = 0 |
| 1 | 1 | 1^3 = 1 |
| 2 | 8 | 2^3 = 8 |
| 3 | 7 | 3^3 = 27 |
| 4 | 4 | 4^3 = 64 |
| 5 | 5 | 5^3 = 125 |
| 6 | 6 | 6^3 = 216 |
| 7 | 3 | 7^3 = 343 |
| 8 | 2 | 8^3 = 512 |
| 9 | 9 | 9^3 = 729 |
The last digit of a perfect cube uniquely identifies the last digit of its cube root. Use this for rapid mental estimation.
Frequently asked questions
Can you take the cube root of a negative number?
Yes. Every real number has exactly one real cube root, and a negative number has a negative cube root. The cube root of -27 is -3, because (-3) x (-3) x (-3) = -27. This differs from square roots, where negatives produce no real result. This calculator handles negatives automatically for any odd root degree.
What is the difference between a square root and a cube root?
A square root (n = 2) finds y such that y^2 = x; a cube root (n = 3) finds y such that y^3 = x. Square roots of negatives are not real, but cube roots of negatives are. A positive number has two square roots (positive and negative) but exactly one real cube root.
How do I check that a cube root is correct?
Multiply the result by itself three times. If the product equals your original number the answer is right. This calculator shows that verification automatically in the "Verification" output row, and the show-your-work panel explains each step.
How does the reverse-solve mode work?
Switch to "Reverse solve" mode, enter the root value y and the degree n. The calculator returns x = y^n. For instance, entering y = 5 and n = 3 returns x = 125, because 5 is the cube root of 125. This is handy when you know the root and need the original number.
How do I find a fourth root or fifth root?
Change the "Root degree (n)" input to 4 or 5 (or any integer from 2 to 100). The formula is always x^(1/n). For even degrees the input must be non-negative for a real result. For odd degrees negatives are allowed and the root carries the same sign as the input.
What is a perfect cube?
A perfect cube is an integer whose cube root is also an integer: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, and so on. The calculator labels the result "Yes, y^3 = x" when it detects a perfect cube. Memorizing the first ten perfect cubes lets you estimate other cube roots by bracketing them between two known values.
Can every real number have three cube roots?
In the real numbers, every real number has exactly one real cube root. In the complex numbers, every non-zero number has three cube roots (one real and two complex conjugate pairs for real inputs). This calculator returns only the real cube root, which is standard for practical use.