Cube Roots

Last Updated : 13 Feb, 2026

A cube root of a number 'a' is a value b such that when multiplied by itself three times (i.e., b \times b \times b), it equals a. Mathematically, this is expressed as:

b3 = a, where a is the cube of b.

A cube is a number that we get after multiplying a number three times by itself. For example, 27 will be the cube of 3.

cube

Formula

Suppose the cube of a number 'x' is y we can represent this as,

x^3 = y

Now cube root of y is calculated as,

y^{\frac{1}{3}} = x \ \text{or} \ \sqrt[3]{y}=x

Finding Cube Root of a Number

We can easily find cube root of a number using following methods,

1. By Prime Factorization

Prime Factorization is a method through which you can easily determine whether a particular figure represents a perfect cube. If each prime factor can be clubbed together in groups of three, then the number is a perfect cube. 

For example: Let us consider the number 1728. 

Prime factorization of 1728 is

1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3

1728 = 23 × 23 × 33

Here, all numbers can be clubbed into groups of three. So we can definitely say that 1728 is a perfect cube.

In fact, the cube root of 1728 is a product of numbers taken one from each group i.e.,  2 × 2 × 3 = 12. 

Example: Find cube root of 216.

216 = 2 × 2 × 2 × 3 × 3 × 3

216 = 23 × 33 = 63

Hence, cube root of 216 is 6.

2. Estimation Method

In this method, we can estimate cube of a perfect cube number using some rule. The steps to estimate cube of perfect cube number are as follows:

Step 1: Take any cube number say 117649 and start making groups of three starting from the rightmost digit of the number. 

So 117649 has two groups, and the first group is (649) and the second group is (117).

Step 2: The unit’s digit of the first group (649) will decide the unit digit of the cube root. Since 649 ends with 9, the cube root’s unit digit is 9.

Note: We can use the following table for finding the unit digit of cube root,

Unit digit of Cube Root123456789
Unit digit of its Cube187456329

Step 3: Find the cube of numbers between which the second group lies. The other group is 117. 

We know that 403= 64000 i.e., second group for cube of 40 is 64, and 503= 125000 i.e., second group for 50 is 125. As 64 < 117 < 125. Thus, the ten's digit of the required number is either 4 or 5 and 50 is the least number with 5 as ten's digit. Thus, 4 is the ten's digit of the given number.

So, 49 is cube root of 117649.

Note: For help with second group we can use the following table,

Number0102030405060708090
Cube0100080002700064000125000216000343000512000729000

Example: Estimate Cube root of number 357911.

Let's take another cube number, say 175616.

  • Step 1: Starting from the rightmost digit, group the digits in threes. So, the first group is (616) and the second group is (175).
  • Step 2: The unit digit of the first group (616) is 6, which corresponds to the unit digit of the cube root 6.
  • Step 3: Find the cubes of numbers between which the second group lies. We know that 43³ = 79507 and 44³ = 85184. Since 175 is between 79507 and 85184, the tens digit of the required number is 4.

Therefore, cube root of 175616 is 46.

Properties of Cube Roots

  • The cube root is denoted as \sqrt[3]{a}
  • Every real number has one real cube root.
  • The cube root of a negative number is also negative, e.g., \sqrt[3]{-8} = -2
  • Product Property: \sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}
  • Quotient Property: \sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}
  • Exponent Property: \sqrt[3]{a^3} = a for all real numbers a
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