Mathématiques
What Is a Cube Root?
∛64 is 4 and ∛(−27) is −3. Cube roots differ from square roots in exactly two ways: negatives are allowed, and there is only ever one real answer.
Cube root
The cube root of a number is the value that gives that number when multiplied by itself three times; ∛64 = 4 because 4 × 4 × 4 = 64.
- Où les élèves le rencontrent
- Grade 7 to 9, taught with square roots once cube numbers are known, and put to work in volume questions where the edge of a cube has to be recovered from its capacity.
La réponse en bref
The cube root of a number is the value that gives that number when multiplied by itself three times, so ∛64 = 4 because 4 × 4 × 4 = 64. Unlike square roots, negatives are allowed: ∛(−27) = −3. And a cube root has exactly one real answer, never two.
Un exemple
A cube of volume 512 cm³ has edges of ∛512 = 8 cm
This is where cube roots stop being an exercise. Volume of a cube is edge × edge × edge, so recovering the edge from the volume is precisely a cube root. 8 × 8 × 8 is 512, so each edge is 8 cm — and the units follow the same logic, cm³ going back to cm.
The first ten cubes, which cover most questions
Cube root questions at this level draw almost entirely on this list, so recognising the numbers in it is most of the skill. 8, 27, 64, 125 and 1000 are the ones that appear again and again in volume and scale-factor work.
Two further cubes earn their place because unit conversion runs on them: 20³ is 8000, and 100³ is a million, which is why one cubic metre holds a million cubic centimetres rather than the hundred a first guess suggests.
- 1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125
- 6³ = 216, 7³ = 343, 8³ = 512, 9³ = 729, 10³ = 1000
Why a negative has a cube root but not a square root
Count the minus signs. Squaring uses two negative factors, and two negatives multiply to a positive, so nothing real can square to a negative number. Cubing uses three, and the third one puts the sign back: (−3) × (−3) is 9, and 9 × (−3) is −27.
So ∛(−27) is −3, and it is an ordinary answer rather than a special case. Any odd power keeps the sign of the number it started from, which is the general rule behind this and worth stating once, because it also settles what happens with fifth roots and every other odd root later on.
One real answer, so no ± is needed
x² = 25 has two solutions because both 5 and −5 square to 25. x³ = 27 has one, because 3 cubes to 27 and −3 cubes to −27, so only one candidate works. Every real number has exactly one real cube root, positive or negative.
The practical consequence is small but easy to get wrong in both directions: writing ± after taking a cube root invents a solution that does not exist, while forgetting it after a square root loses one that does. The two operations look alike on the page and behave differently.
Questions fréquentes
What is the cube root of 27?
3, because 3 × 3 × 3 = 27. It is worth checking such answers by multiplying back rather than trusting recall, since 3³ and 3² are easy to confuse under time pressure — 9 and 27 both look plausible at a glance.
Can you take the cube root of a negative number?
Yes, and the answer is negative. ∛(−125) is −5, since (−5) × (−5) × (−5) = −125. This is the clearest difference from square roots, where a negative input has no real answer at all. Three negative factors leave the result negative; two do not.
Do I need a ± sign when taking a cube root?
No. A cube root has one real value, so x³ = 64 gives x = 4 and nothing else. The ± belongs with even roots, where two values square to the same positive number. Adding it to a cube root produces an answer that does not satisfy the original equation.
How do I find a cube root on a calculator?
Use the ∛ key if there is one, or raise the number to the power of one third. On most scientific calculators the cube root sits above another key and needs the shift function. If the number is one of the first ten cubes, recognising it is quicker than either.
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