Cubes and Cube Roots

Cube and Cube Roots are fundamental concepts in algebra and it is introduced in the early classes. The multiplication of a number to itself gives rise to a square and then if we multiply the number by its square the result becomes a cube and the inverse of the cube is the cube root which we will study in this article.

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

Cube and Cube Roots

The cube of a number is the result of multiplying that number by itself three times. The cube of a number n is denoted as n³. For example:

• The cube of 2 is 2³ = 2×2×2 = 8.
• The cube of -3 is (−3)³ = −3 × −3 × −3 = −27.

Cubes are closely related to the geometric concept of a cube, where the volume of a cube (a three-dimensional object with all sides of equal length) is calculated as s³, where s is the length of one side.

The cube root of a number x is a number a that, when cubed, equals x. It is denoted as ​. For example:

• The cube root of 8 is 2​ because 2³ = 8.
• The cube root of -27 is -3 because (−3)³ = −27.

Finding the cube root is the inverse operation of finding the cube. Cube roots can be both positive and negative because cubing a negative number results in a negative product.

What is Cube?

Physically a cube is a solid figure that has all its sides equal. In mathematical terms, the cube of a number is the result of multiplying a whole number by itself twice, i.e., the whole number is used three times just like the sides of a cube. For example: 

• Cube of 1 = 1³ = 1 × 1 × 1 = 1,
• Cube of 2 = 2³ = 2 × 2 × 2 = 8,
• Cube of 3 = 3³ = 3 × 3 × 3 = 27 . . . and so on.

Cubes 1 to 20

The cube of natural numbers 1 to 20 is discussed in the table below,

Properties of Cube of Numbers

There are various properties of the cube of numbers, some of which are as follows:

• Cubes of all odd numbers are odd.
• Cubes of all even numbers are even.
• Cubes of numbers ending with 2 will end in 8. Similarly, cubes of numbers ending with 8 always end with 2.
• Cubes of numbers ending with 3 will end in 7. Similarly, cubes of numbers ending with 7 always end with 3.
• Cubes of numbers ending with 4, 5, 6, and 9 will end in 4, 5, 6, and 9 respectively.

Representation of Cube Numbers

Cube Numbers can be represented as the volume of cubes as volume is also a three-dimensional quantity and calculated by the multiplication of side length by itself three times. The following illustration shows the representation of cube numbers:

Cube of Negative Numbers

Similar to positive integers we can find the cube of negative integers as well. It is the same for negative numbers as well. To find the cube of a negative number we just multiply the negative number three times by itself. As a product of two negative numbers is positive and the product of negative and positive numbers is negative. Thus, a cube of a negative number is always negative. 

For example, a cube of -2 is nothing but (-2)×(-2)×(-2) = 4×(-2) = -8.

• Cube of (-3) = -27
• Cube of (-4) = -64
• Cube of (-5) = -125 . . . and so on,

Cube of Fractions

A cube of any number is defined as the product of itself three times or the number raised to the third power. For fractions, cubes can be calculated by individually calculating the cubes of the numerator and denominator in the fraction. For example, a cube of (3/2) is simply 3³/2³ = 27/8 [As the cube of 3 is 27 and the cube of 2 is 8]. Some more examples of cubes of fractions are as follows:

• Cube of (-1/3) = -1/27
• Cube of (4/5) = 64/125
• Cube of (7/5) = 343/125

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.

Units Digit of Number	Units Digit of its Cube
1	1
2	8
3	7
4	4
5	5
6	6
7	3
8	2
9	9

Perfect Cubes

A perfect cube is a positive integer equal to some other positive integer raised to the third power. In other words, when we multiply an integer by itself the resulting number is called a perfect cube. For example, 125 is a perfect cube since 125 = 5 × 5 × 5 = 5³. However, 121 is not a perfect cube because there is no integer n such that n³ = 121 = 11². Some other examples of perfect cubes are 1, 8, 27, 64, 125, 216, 343, . . . , etc.

What is Cube Root?

Cube root of a number is a value that, when multiplied three times, gives that number. For example, 3 × 3 × 3 = 27, so cube root of 27 is 3.

Symbol of Cube Root

The cube root of any number in mathematics is represented as [Tex]\bold{\sqrt[3]{x}} [/Tex].

• [Tex]\sqrt[3]{27} = 3[/Tex]
• [Tex]\sqrt[3]{125} = 5[/Tex]
• [Tex]\sqrt[3]{\left(\frac{1}{27}\right)} = \frac{1}{3}[/Tex]

Cube Root Formula

Cube root formula is formula that is used to find cube root of any substance. Suppose 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

Image added below represent 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.

Cube Root of a Cube Number using Estimation

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.

Cube and Cube Roots Class 8

In mathematics, particularly in Class 8, students learn about cubes and cube roots as part of their number theory or algebra curriculum. Understanding these concepts involves exploring the relationships between numbers, their cubes, and their cube roots.

Cube and Cube Roots Class 8 Worksheet

1. Calculate the cube of 7.

2. Find the cube root of 729.

3. Determine whether 216 is a perfect cube. If it is, identify which number’s cube it is.

4. A small container is shaped like a cube and has a volume of 64 cubic inches. What is the length of one side of the container?

5. Simplify the expression 34333343​ and explain your process.

6. A metal cube weighs 125 grams and has a density of 5 grams per cubic centimeter. Find the length of a side of the cube.

FAQs on Cube and Cube Roots

What is Difference between 3D Cube and Cube Numbers?

3D Cube is solid and a physical object as cube numbers are the abstract concept which is a number we get after raising any number to it’s third power.

What is Cube of 3?

Cube of 3 is 27.

What is Cube of 4?

Cube of 4 is 64.

How to Calculate Cube Root of a Number?

We can find cube root of a perfect cube number using prime factorization method or we can estimate cube root using steps explained in this same article.

What is Cube Root of 27?

A cube of 3 is 27, therefore cube root of 27 is 3.

What is Cube Root of 125?

A cube of 5 is 125, therefore cube root of 125 is 5.

What is relationship between Cubes and Cube Roots?

Cube and Cube Roots are inverse operations that undo, effect of each other on any number. In other words, if we take any number and take it’s third power to find its cube and then find the cube root of the result. We will end up with the same number we have taken at first.