Cube Roots

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.

Cube is a number that we get after multiplying a number 3 times by itself.

For example, 125 will be the cube of 5. While cube root of a number is that number which is multiplied 3 times to get the original number. For example, the cube root of 125 is 5 if we multiply 5 three times will be 125.

In this article, we will learn about cube roots and also learn about the methods to find cube roots of a number.

Cube Root Formula

The cube root formula is a formula that is used to find cube root of any substance. Suppose the cube of a number is x is y we can represent this as,

x³ = y

Now cube root of y is calculated as,

y^1/3 = x

The image added below represents the same.

How to Find Cube Root of a Number?

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

• By Prime Factorization
• By Estimation Method

Now, let's learn about them in detail.

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 1: Find cube root of 216.

216 = 2 × 2 × 2 × 3 × 3 × 3

216 = 2³ × 3³ = 6³

Hence, cube root of 216 is 6.

Example 2: Find cube root of 5832.

5832 = 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3 

5832 = 2³ × 3³ × 3³ = 18³

Hence, cube root of 5832 is 18.

By 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 Root	1	2	3	4	5	6	7	8	9
Unit digit of its Cube	1	8	7	4	5	6	3	2	9

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 requred 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,

Number	0	10	20	30	40	50	60	70	80	90
Cube	0	1000	8000	27000	64000	125000	216000	343000	512000	729000

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.

Hardy-Ramanujan Numbers

Numbers like 1729, 4104,13832, etc. are known as Hardy - Ramanujan Numbers because they can be expressed as the sum of two cubes in two different ways. A number n is said to be Hardy-Ramanujan Number if

n = a³ + b³ = c³ + d³

where,

• a, b, c, and d are all distinct positive integers

First four Hardy-Ramanujan numbers are:

• 1729 = 1³ + 12³ = 9³ + 10³
• 4104 = 2³ + 16³ = 9³ + 15³
• 13832 = 2³ + 24³ = 18³ + 20³
• 20683 = 10³ + 27³ = 19³ + 24³

If we consider negative integers as well, then 91 becomes the smallest Hardy-Ramanujan Number as it can be expressed as follows:

91 = 6³ + (-5)³ = 4³ + 3³

Note: Number 1729 is also sometimes referred to as the Taxicab Number as it was the number of taxi taken by Dr. Hardy while going to meet Ramanujan at hospital.

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} = aor all real numbers aa

Unit Digits in Cube Numbers

• If a number is odd, its cube number unit digit is also odd.
• And similarly, If a number is even, its cube number unit digit is also even.

Following is a table that shows the unit digit of a number and the unit digit of the cube of that number.

Related Articles:

• Cubes and Cube Roots Class 8 Notes
• Cubes and Cube Roots NCERT Solution Class 8
• Cube Root of 1728
• Cube Root of 1000
• Cube Root of 13
• Cube Root of 5
• Cube Root of 125
• Cube Root of 216
• Cube Root 1 to 20
• Cube Root 1 to 30

Cube Roots Worksheet

You can download this worksheet from below with answers: