[Maths Class Notes] on Perfect Cube Of Numbers Pdf for Exam

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Table of Content 

• What is a perfect number?

• What is a square?

• What is a perfect square?

• What is a cube?

• Interrelation of cubes and algebra 

• History of Algebra

• Uses of Algebra 

• Properties of a perfect cube

• Perfect cube of First 50 natural numbers 

• Solved examples 

• Square root

• Cube root 

• Symbol of cube root

• Key points from the chapter

• Frequently asked questions
  

What are Perfect Numbers?

It is said that the Egyptians were the ones who discovered the perfect number. However, it was the curiosity of the Greeks which made them study the concept of the perfect number and bring it forward for the world to study. 

So, perfect numbers are the positive integers that are equal to the sum of their factors except for the number itself. In other words, perfect numbers are the positive integers which are the sum of their proper divisors. 

For example – The smallest perfect number is 6, which is the sum of its proper divisors: 1, 2 and 3

What is a Square?

A square is a flat shape with four equal sides having a 900 angle in every interior.

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Perfect Square

A square of a number is acquired by multiplying a specific number to itself. We can say that an ideal square is a number that is obtained by the increase of a number without any other number.

For instance,

4 = 2 x 2

25 = 5 x 5

Properties of Perfect Squares are:

1. Perfect squares do not have a unit digit of 2, 3, or 7.

2. All even square numbers are divisible by 4.

3. The difference of 2 odd squares is a multiple of 8.
   

What is a Cube?

A Cube is a three-dimensional figure whose edges are of the same length. The volume of a cuboid is given by the product of its measurements.

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Perfect Cube

The idea of a Perfect Cube is a build-up to the idea of a perfect square. The thing that matters is that in cubes, we increase a number thrice. The perfect cube of a number can be characterized as the number obtained by multiplying a number by itself two times.

For example, 8 is known as a perfect cube number because it is obtained by multiplying 2 thrice;

i.e. 8 = 2 x 2 x 2.

We may consider a perfect cube as a cube of each side measuring x units and the volume of such a cube will be x3 cubic units.

Numbers like 23 = 8, 33 = 27, 43 = 64, then the numbers 8, 27, 64 are termed as perfect cubes. 

Therefore, a number is defined as a perfect cube, if it is a cube of another natural number.  Suppose that a and b are two different natural numbers, then a is said to be a perfect cube of b if  a = b x b x b i.e. a = b3

Let’s go ahead and understand more about perfect cubes.

Cubes are an Important Part of Algebra.

Algebra is the section of mathematics that deals with symbols and rules for control of these symbols. In elementary algebra, the symbols are known as variables that do not have fixed values. These variables form equations that then describe the relationship between them.

Uses of Algebra

• Algebra helps you in your career

• It is a powerful tool that generalizes situations

• It helps you think logically

• It is very important for modern technology
  

History of Cubes

Calculation of the cubes of large numbers was very common in many ancient civilizations. By the old Babylonian period (20th to 16th centuries BC), the Mesopotamian Mathematicians invented cuneiform tablets with tables for calculating cubes and cube roots. The ancient Greek Diophantus also knew about the Cubic equations. The hero of Alexandria designed a method for calculating cube roots in the 1st Century. The methods for solving cubic equations and calculating cube roots are mentioned in the nine Chapters of Mathematical Art. Our own respected mathematician Aryabhata wrote an Explanation of cubes. In 2010 Alberto Zanoni found a faster squaring and multiplying way to calculate the cube of a long integer in a certain range.

Properties of Perfect Cubes

1. The cube value of an even number will always be an even number.

For instance,

43 = 4 x 4 x 4 = 64

23 = 2 x 2 x 2 = 8

2. The cube value of an odd number will always be an odd number.

For instance,

33 = 3 x 3 x 3 = 27

73 = 7 x 7 x 7 = 343

3. The cubes of numbers which end in 0, 1, 4, 5, 6, 9, often give the result in the same digits.

For instance,

103 = 1000 ends in 0

113 = 1331 ends in 1

143 = 2744 ends in 4

53 = 125 ends in 5

163 = 4096 ends in 6

193 = 6859 ends in 9

4. One of the most vital properties is that the sum of the first and successive natural number’s cubes is the same as the whole square of their sum.

13+23+33+43…+n3 = (1+2+3+…+n)2

Have a look at following example:

13+23+33 = 1 + 8 + 27 = 36 and (1+2+3)2 = (6)2 = 36

How to find whether a number is a perfect cube or not?

To find that whether a number is a perfect square or not follow the steps mentioned below:

• Check if the given number is a natural number, if yes then keep going.

• Find the prime factors of the number.

• Group the factors in triplets, in such a way that each triplet has three same factors.

• If no factor is left after grouping then the number is a perfect cube.
  
  

Examples of Perfect Cube Concept.

1. Is 512 a Perfect Cube? Find its Cube Root.

Solution:

By factoring 512 into prime factors, we get

512 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2

After grouping the numbers in triplets, we get

512 = (2 x 2 x 2) x (2 x 2 x 2) x (2 x 2 x 2)

Hence 512 is a perfect cube as no number is leftover.

Taking one number from each group,

512 = (2)x(2)x(2) = 8 

Therefore, 512 is the perfect cube of 8.

2. Calculate Whether 9261 is a Perfect Cube.

Solution:

After factoring 9261 into prime factors, we get

9261 = 3 x 3 x 3 x 7 x 7 x 7

Group these numbers into triplets

9261 = (3 x 3 x 3) x (7 x 7 x 7)

No number is leftover, so 9261 is a perfect cube.

Taking one number from each group,

9261 =  (3) x (7) = 21

Hence, 9261 is the perfect cube of 21. 

3. Find the Smallest Number by Which 1323 Shall be Multiplied in Order to Build a Perfect Cube?

Solution:

Factoring 1323 into prime factors, we get

1323 = 3 x 3 x 3 x 7 x 7

Now to group these factors into triplets, one more 7 is needed. Hence, 7 is the smallest number which should be multiplied to 1323 to make it a perfect cube.

4. What Would be the Smallest Number by Which the Number 1375 Should be Divided to Make the Quotient a Perfect Cube? Find the Quotient and its Cube Root.

Solution:

We have

1375 = 11 x 5 x 5 x 5

1375 = 11 x (5 x 5 x 5)

11 is leftover which means that if we divide 1375 by 11, then the quotient will be a perfect cube.

Dividing 1375 by 11, we get

137511/11 = 5×5×5

= 125

The number is 125 and 5 is its cube root.

Perfect Cubes of First 50 Natural Numbers are as Follows:

1. 1 = 1 x 1 x 1 = 13

2. 8 = 2 x 2 x 2= 23

3. 27 = 3 x 3 x 3= 33

4. 64 = 4 x 4 x 4 = 43

5. 125 = 5 x 5 x 5 = 53

6. 216 = 6 x 6 x 6= 63

7. 343 = 7 × 7 × 7 = 73

8. 512 = 8 × 8 × 8 = 83

9. 729 = 9 × 9 × 9 = 93

10. 1000 = 10 × 10 × 10 = 103

11. 1331 = 11 × 11 × 11 = 113

12. 1728 = 12 × 12 × 12 = 123

13. 2197 = 13 × 13 × 13 = 133

14. 2744 = 14 × 14 × 14 = 143

15. 3375 = 15 × 15 × 15 = 153

16. 4096 = 16 × 16 × 16 = 163

17. 4913 = 17 × 17 × 17 = 173

18. 5832 = 18 × 18 × 18 = 183

19. 6859 = 19 × 19 × 19 = 193

20. 8000 = 20 × 20 × 20 = 203

21. 9261 = 21 × 21 × 21 = 213

22. 10648 = 22 × 22 × 22 = 223

23. 12167 = 23 × 23 × 23 = 233

24. 13824 = 24 × 24 × 24 = 243

25. 15625 = 25 × 25 × 25 = 253

26. 17576 = 26 × 26 × 26 = 263

27. 19683 = 27 × 27 × 27 = 273

28. 21952 = 28 × 28 × 28 = 283

29. 24389 = 29 × 29 × 29 = 293

30. 27000 = 30 × 30 × 30 = 303

31. 29791 = 31 × 31 × 31 = 313

32. 32768 = 32 × 32 × 32 = 323

33. 35937 = 33 × 33 × 33 = 333

34. 39304 = 34 × 34 × 34 = 343

35. 42875 = 35 × 35 × 35 = 353

36. 46656 = 36 × 36 × 36 = 363

37. 50653 = 37 × 37 × 37 = 373

38. 54872 = 38 × 38 × 38 = 383

39. 59319 = 39 × 39 × 39 = 393

40. 64000 = 40 × 40 × 40 = 403

41. 68921 = 41 × 41 × 41 = 413

42. 74088 = 42 × 42 × 42 = 423

43. 79507 = 43 × 43 × 43 = 433

44. 85184 = 44 × 44 × 44 = 443

45. 91125 = 45 × 45 × 45 = 453

46. 97336 = 46 × 46 × 46 = 463

47. 103823 = 47 × 47 × 47 = 473

48. 110592 = 48 × 48 × 48 = 483

49. 117649 = 49 × 49 × 49 = 493

50. 125000 = 50 × 50 × 50 = 503

SQUARE ROOTS

A square root is a number which gives a specified quantity when multiplied by itself.

The square of the number 3 is 9 and the square root of 9 is 3. Square and square root are inversely related.

Method to find the square root of a number is as follows: To find the square root of a number, find some number that when multiplied by itself gives the original number. 

For example, to find the square root of 9, find the number that when multiplied by itself gives 9, which is 3, because 3 x 3 = 9.

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CUBE ROOTS

A cube root is the number that multiplies by itself 3 times to create a cubic value. Cube roots like the perfect cube equations were created and used for calculation by the Babylonian mathematicians.

The cube of the number 2 is 8 and the cube root of 8 is 2. Cubes and cube roots are inversely related.

Methods to find the cube root of a number are as follows: In order to find the cube root of a number, find some number that when multiplied by itself twice gives you the original number. For instance, to find the cube root of 8, find the number that when multiplied by itself twice gives 8, which is 2, because 2 x 2 x 2 = 8.

Key Points From the Chapter –

• All the perfect numbers are even so it is not clear whether odd perfect numbers exist or not

• Perfect numbers end with 6 or 8 

• Sum of all the divisors of a perfect number gives the as twice the perfect number

• Perfect numbers are the positive integers as a result of the sum of their proper divisors

• The smallest perfect number is 6