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.
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.
Sky is the Limit
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1. What do you mean by a perfect cube?
Ans. A perfect cube is nothing but a number that is derived cubing a whole number. For instance, 27 is a perfect cube, since you can derive the same from the equation 33. However, the number 45 is not a perfect cube, since its cube root is not a whole number, but instead an integer. One should remember that even though 45 is not an example of a perfect cube, you can still derive its cube root in decimals. By long division method, one can ascertain that its cube root would be approximately 3.5568. In fact, you can determine the cube root for any number irrespective of whether it is a perfect cube.
2. What is the simplest way to determine whether the perfect cube of a number would be even or odd?
Ans. You should understand that odd numbers would always have odd perfect cube figures. Similarly, any even number will result in even cubes. For instance, the cube of 8, which is an even number, would be 512, another even number. In the same way, the cube of an odd number, 7, is 343, which is another odd number.
3. What is an interesting factor for cubes of numbers ending with 3 and 7?
Ans. It is interesting to note that all numbers ending with 3 will have a cube figure that ends with the number 7. For instance, the cube of 3 is 27.
Similarly, any number ending with the number 7 will have a cube ending with number 3. For instance, the cube of the number 7 is 343. This rule can help in determining cubes and cube roots faster for various whole numbers.
4. What is their prime factorisation?
Ans. Prime factorisation of any number can be completed by determining the prime factors of any figure. Next, you can club these prime factors in pairs of three. If all prime factors can be paired in sets of three, then the starting number is a perfect cube. This is a crucial part of understanding cubes and cube roots.
To help you understand, let us take an example where prime factorisation can help you determine whether a specific number is a perfect cube. Let us assume 729 to be the number. We can alternatively write 729 in the following manner
3 x 3 x 3 x 3 x 3 x 3 = 729
33 x 33 = 729
Therefore, we can say that 729 is a perfect cube, since each prime factor can be paired with three others of its kinds.