The cube root of a number is the value that, used as a factor three times, gives that number. It is written 3√x. Because 4 × 4 × 4 = 64, 3√64 = 4. Unlike square roots, cube roots of negative numbers are real numbers: 3√−27 = −3.
Cubing a number gives the volume of a cube with that edge; the cube root goes backward from the volume to the edge. Negative numbers work too, because three negative factors give a negative product.
Formal definition and notation
For any real numbera, the cube root 3√a is the one real number b with b3 = a. Every real number has exactly one real cube root, with the same sign as the number, and 3√0 = 0. It is a radical with index 3 and can also be written with an exponent as a13. Taking a cube root undoes cubing: 3√a3 = a and (3√a)3 = a for every real a. Perfect cubes such as 8, 27 and 64 have integer cube roots; the cube root of a whole number that is not a perfect cube is an irrational number, such as 3√2 ≈ 1.26.
Cube roots grow slowly: multiplying a number by 1,000 only multiplies its cube root by 10.
Definition of cube root
Definition of cube root
∛a = b exactly when b3 = a
a
any real number: positive, negative or zero
b
its cube root, which has the same sign as a
For a cube with volume V, each edge is s = 3√V. A cube with volume 1,000 cubic centimeters has edges of 3√1000 = 10 cm.
Try it: cube a number
Interactive
Keep the exponent at 3 and change the base. The result is a perfect cube, and the base is its cube root. Try negative bases such as −2 and −5 to see negative cubes.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Cube Root examples, step by step
Example 1: A perfect cube
Warm-up
Find 3√125.
Ask which number used three times as a factor gives 125.
5 × 5 = 25, and 25 × 5 = 125.
Answer:3√125 = 5.
Example 2: A negative and a decimal
Core
Find 3√−216 and 3√0.008.
63 = 216, so (−6)3 = −216. That gives 3√−216 = −6.
Write 0.008 as 81000. Then 3√8 = 2 and 3√1000 = 10.
A square root uses a number as a factor twice: √64 = 8 because 8 × 8 = 64. A cube root uses it three times: 3√64 = 4. Negative numbers have real cube roots but no real square roots.
Cube Rootvs.Perfect cube
A perfect cube is the result of cubing an integer, such as 27 or −125. The cube root is the number you started with: 3 or −5.
Cube Rootvs.Cubed (third power)
Cubing goes forward: 33 = 27. The cube root goes backward: 3√27 = 3. They undo each other.
Quick check: 3 questions
Question 1 of 3
What is 3√343?
Show answer and explanation
Answer: C.7 × 7 × 7 = 343, so the cube root is 7. 49 is 72, 114.3 divides by 3, and 18.5 is close to the square root of 343.
Question 2 of 3
What is 3√−64?
Show answer and explanation
Answer: A.(−4)3 = −64, so 3√−64 = −4. 4 has the wrong sign, odd roots of negative numbers are real, and −8 comes from 8 × 8 = 64, which is square root thinking.
Question 3 of 3
Between which two whole numbers is 3√100?
Show answer and explanation
Answer: D.43 = 64 and 53 = 125, and 100 is between them, so 3√100 is between 4 and 5 (about 4.64). 9 and 10 mixes it up with √100 = 10, 3 and 4 only reach 64, and 33 and 34 come from dividing by 3.
Cube Root: frequently asked questions
Can you take the cube root of a negative number?
Yes. A negative number times itself three times is negative, so every negative number has a negative cube root: 3√−125 = −5 because (−5)3 = −125. This is different from square roots, where √−25 is not a real number, since any real number squared is zero or positive.
How do you find a cube root without a calculator?
Learn the perfect cubes from 1 to 1,000. If the number is a perfect cube, read off the root. If it is not, find the two perfect cubes around it to estimate: 40 is between 27 and 64, so 3√40 is between 3 and 4. You can also factor out a perfect cube: 3√40 = 23√5.
Is the cube root of 2 a rational number?
No. 2 is not a perfect cube, so 3√2 ≈ 1.2599 is irrational: its decimal never ends or repeats, and it cannot be written as a fraction of integers. The same is true for the cube root of any whole number that is not a perfect cube, such as 3√10 or 3√100.
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