How to Find Cubes and Cube Roots Easily
Class 8 students need to learn all chapters in the mathematics curriculum with proper care. However, you must focus more intensely on the cubes and cube roots chapter. This lesson can help you throughout your academic career, even if you choose to pursue competitive examinations in the future.
With a proper understanding of cubes and cube roots class 8 curriculum, you can solve almost any questions that arise from this chapter. In the following lesson, students will learn about the Hardy-Ramanujan Number, prime factors and other useful factors regarding cubes.
What is The Hardy-Ramanujan Number?
Hardy-Ramanujan number refers to any figure, which can be expressed by the summation of two cubes. For example, 1729 is not a perfect cube but you can express the same as 1728 + 1 or 123 + 13. The same number can also be expressed as 1000 + 729, which is also 103 + 93. 1729 is the smallest Hardy-Ramanujan Number. Other numbers in this series include 4104, 13832 and many more.
Properties of a Cube Number
When studying chapter 7 cubes and cube roots, one should pay close attention to the various characteristics of cubes. These properties include the following factors –
Cubes of all odd numbers are odd.
Cubes of all even numbers are even.
Cubes of numbers ending with two will end in eight. Similarly, cubes of a number ending with eight always end with two.
Cube of a number ending with 3 will always end with 7. Also, the cube of a number ending with 7 will always end with 3.
The sum of cubes of first ‘n’ natural numbers will always be equal to the square of their sums.
Assessing Prime Factorisation in CBSE class 8 Maths Cubes and Cube Roots
Prime factorisation 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 in question is a perfect cube. To understand how prime factorisation method works, let us take the help of an example.
Consider the number 500 to be divided using prime factorisation. Therefore, we can say,
500 = 2 x 2 x 5 x 5 x 5 = 22 x 53
Now, let us consider the number 1728.
1728 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 = 23 x 23 x 33
Here, all numbers form cubes, which is why, we can say definitively that 1728 is a perfect cube number. In fact, its cube root is 12. Prime factorisation is one of the most important aspects of cubes and cube roots. Therefore, ensure you have no doubts regarding this topic.
True or False
Q. Any cube of a natural number with 3 as its multiple will also have 27 as a multiple.
Ans. True.
Cube Root Estimation
Let us assume a number, say 205379. Start clubbing three numbers from the right hand side. So, the first group will have 379, while the second group will have 205. Since the unit place of the first group is 9, we can say that the cube root’s unit place will also be 9.
Next, take the second group number and determine the closest perfect cubes to it. The closest perfect cubes to 205 are 125, which is 53, and 216, which is 63. Assume the lower cube root number as the first digit for the starting number. Thus, with this method, you can arrive at the result, 59, which is indeed the cube root for 205379.
Keep practicing cubes and cube roots exercises to gain a better understanding of this topic.
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FAQs on Cubes and Cube Roots Explained
1. What is the difference between a cube and a perfect cube in mathematics?
A cube is a number obtained by multiplying an integer by itself three times. For example, the cube of 2 is 2 × 2 × 2 = 8. A perfect cube is the result of this operation. Therefore, 8 is a perfect cube. While any integer can be cubed, a number is only called a perfect cube if its cube root is an integer. For instance, 10 is not a perfect cube because its cube root (approx. 2.154) is not an integer.
2. How do you find the cube root of a number using the prime factorisation method?
The prime factorisation method is a fundamental technique for finding the cube root of a perfect cube. The steps are as follows:
First, find all the prime factors of the given number.
Next, group these prime factors into identical triplets (groups of three).
For the number to be a perfect cube, every prime factor must form a triplet.
Finally, take one factor from each triplet and multiply them together. The resulting product is the cube root of the original number.
For example, for 216, the prime factors are 2 × 2 × 2 × 3 × 3 × 3. This gives us one triplet of 2s and one triplet of 3s. Taking one from each, we get 2 × 3 = 6. So, the cube root of 216 is 6.
3. What is the pattern for the last digit of a number and the last digit of its cube?
There is a consistent pattern between the unit digit of a number and its cube, which helps in estimation. The patterns are:
Numbers ending in 0, 1, 4, 5, 6, or 9 have cubes that also end in 0, 1, 4, 5, 6, or 9, respectively.
Numbers ending in 2 have cubes that end in 8 (e.g., 2³ = 8, 12³ = 1728).
Numbers ending in 8 have cubes that end in 2 (e.g., 8³ = 512).
Numbers ending in 3 have cubes that end in 7 (e.g., 3³ = 27, 13³ = 2197).
Numbers ending in 7 have cubes that end in 3 (e.g., 7³ = 343).
4. How can you determine if the cube of a number will be even or odd without calculating it?
You can determine this by looking at the original number itself. The cube of an even number is always even, and the cube of an odd number is always odd. For example, 4 is an even number, and its cube, 64, is also even. Similarly, 5 is an odd number, and its cube, 125, is also odd. This property holds true for all integers.
5. Why must the prime factors of a perfect cube appear in groups of three?
The definition of a cube root is a number 'a' which, when multiplied by itself three times (a × a × a or a³), gives the original number 'N'. When we perform prime factorisation on 'N', we are breaking it down into its fundamental building blocks. For 'N' to be a perfect cube, its building blocks (prime factors) must also be groupable into three identical sets. Each set represents the prime factors of the cube root 'a'. Therefore, having each prime factor appear in triplets ensures that the factors can be distributed evenly three times, forming a valid integer cube root.
6. How does the concept of a cube root apply to negative numbers?
Unlike square roots, negative numbers can have real cube roots. This is because multiplying a negative number by itself three times results in a negative number. For example, the cube root of -27 is -3, because (-3) × (-3) × (-3) = 9 × (-3) = -27. Therefore, the cube of any negative integer is always a negative integer, and the cube root of a negative perfect cube is always a negative integer.
7. What is the main difference between finding a square root and a cube root?
The main difference lies in the number of identical factors you are looking for. To find a square root, you look for a number that, when multiplied by itself (in pairs), gives the original number. For a cube root, you look for a number that, when multiplied by itself three times (in triplets), gives the original number. This also affects their properties; for example, a negative number does not have a real square root, but it always has a real cube root.
8. What is the estimation method for finding the cube root of a large perfect cube?
The estimation method, as per the NCERT syllabus for the academic year 2025-26, is a quick way to find the cube root of large perfect cubes. The steps are:
Split the number into groups of three digits, starting from the right.
Look at the last digit of the rightmost group. Use the unit digit patterns (e.g., if it's 7, the cube root's unit digit is 3) to find the unit digit of the cube root.
Now, take the leftmost group. Find the two perfect cubes it lies between. For example, if the group is 50, it lies between 3³ (27) and 4³ (64).
Choose the smaller of the two cube roots (in this case, 3). This becomes the tens digit of your answer.
Combine the digits. For a number like 50653, the unit digit is 7 (from the group 653) and the tens digit is 3 (from the group 50). So, the cube root is 37.
9. What is the importance of the Hardy-Ramanujan number 1729?
The number 1729 is known as the Hardy-Ramanujan number and holds a special place in mathematics. Its importance in the context of this chapter is that it is the smallest positive integer that can be expressed as the sum of two different perfect cubes in two different ways:
1729 = 1³ + 12³
1729 = 9³ + 10³
This example illustrates that while basic properties of cubes are simple, the relationships between them can lead to complex and interesting areas of number theory.
