Find the cube root of any number — the value that, when multiplied by itself three times, gives your number. Unlike square roots, negative numbers do have real cube roots.
The cube root is the number that gives your input when multiplied by itself three times.
A negative input gives a negative cube root, because a negative multiplied three times stays negative.
If the result is a whole number, the input is a perfect cube.
Cube root: ∛n = x where x × x × x = n
Also written: n^(1/3)
Finding the cube root of −27.
Number: −27
Look for a number that cubed gives −27
(−3) × (−3) = 9, then 9 × (−3) = −27
−27 is so a perfect cube
Answer: ∛−27 = −3
A cube root calculator finds the number that, multiplied by itself three times, gives your input. It works for negative numbers as well, because a negative number cubed stays negative. It also tells you whether the input is a perfect cube, such as 3,375, whose cube root is exactly 15.
Cube roots answer volume questions. If a cubical box, tank or room holds a certain volume, the cube root gives the length of each edge. Students meet cube roots in Class 8 NCERT through prime factorisation and later in scaling problems, where doubling the volume of a solid increases each length by only about 26%. Engineers and scientists use them when a quantity varies with the cube of a length, as with volume and mass.
1. Note the sign: a negative input gives a negative cube root, so work with the positive value and restore the sign at the end.
2. Try prime factorisation: group equal prime factors in threes and take one from each group. For example, 3375 = 3³ × 5³, so ∛3375 = 3 × 5 = 15.
3. If the number is not a perfect cube, find the two perfect cubes it lies between for a first estimate.
4. Refine an estimate x with Newton's rule: new x = (2x + n ÷ x²) ÷ 3.
5. Repeat until successive estimates agree.
6. Check by cubing the answer.
Squaring a real number can never produce a negative result, because both a positive times a positive and a negative times a negative are positive. Cubing is different. A negative number multiplied by itself gives a positive, and one more multiplication by the negative makes it negative again. So (−4)³ = −64 and ∛(−64) = −4. Every real number has exactly one real cube root, and the cube root function passes smoothly through zero, keeping the sign of its input.
If each edge of a cube is multiplied by k, its volume is multiplied by k³. Turned around, multiplying a volume by V multiplies each edge by ∛V. Doubling a tank's capacity therefore needs edges only ∛2 ≈ 1.26 times longer. This is why a large water tank looks only a little bigger than a small one yet holds far more. It also explains why a 1,000-litre tank, which is exactly 1 cubic metre, can be a cube just 1 m on each side.
One might compute a cube root as n raised to the power 1/3. For negative n that fails on most computers, because fractional powers of negatives are treated as complex and return 'not a number'. This calculator uses a dedicated cube root function that handles negatives correctly. For perfect cubes, the result is checked by rounding and cubing back, so 3,375 is reported as a perfect cube even if the floating-point result has a tiny rounding error.
Nisha has a cubical storage box with a volume of 3,375 cubic centimetres and wants to know the length of each edge before buying a cover.
Answer: Cube root 15; Perfect cube? Yes
Grouping prime factors in pairs, as for square roots, instead of in threes.
Assuming a negative number has no cube root.
Dividing a volume by 3 instead of taking its cube root.
Forgetting to convert litres to cubic centimetres or cubic metres before finding an edge length.
Finding the edge of a cubical box, tank or room from its volume.
Class 8 exercises on cube roots by prime factorisation.
Scaling models, where volumes change by a known factor and lengths must follow.
Working out the side of a cube-shaped die, block or crate from its volume or mass and density.
Why do negative numbers have cube roots but not square roots?
Cubing keeps the sign the same — multiplying three negative numbers together still gives a negative result. Squaring always gives a positive result, so no real number squares to a negative one.
What is a perfect cube?
A number whose cube root is a whole number — like 1, 8, 27, 64, 125, and so on.