Points to Remember- Cubes and Cube Roots Class 8 Notes | EduRev

Mathematics (Maths) Class 8

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Class 8 : Points to Remember- Cubes and Cube Roots Class 8 Notes | EduRev

The document Points to Remember- Cubes and Cube Roots Class 8 Notes | EduRev is a part of the Class 8 Course Mathematics (Maths) Class 8.
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Facts That Matter

  • If we multiply a number by itself three times, the product so obtained is called the perfect cube of that number.
  • There are only 10 perfect cubes from 1 to 1000.
  • Cubes of even numbers are even and those of odd numbers are odd.
  • The cube of a negative number is always negative.
  • If the digit in the one’s place of a number is 0, 1, 4, 5, 6 or 9, then its cube will end in the same digit.
  • If the digit in the one’s place of a number is 2, then the ending digit of its cube will be 8 and vice-versa.
  • If the digit in the one’s place of a number is 3, then the ending digit of its cube will be 7 and vice-versa.
  • If the prime factors of a number cannot be made into groups of 3, it is not a perfect cube.
  • The symbol ∛ denotes the cube root of a number, such as ∛64 = 4.

We Know That
The square of a number is obtained by multiplying the number by itself, i.e.
2 * 2 = 22
4 * 4 = 42
5 * 5 = 52
Finding square root is the inverse operation of finding the square of a number, i.e.
Points to Remember- Cubes and Cube Roots Class 8 Notes | EduRev
A natural number multiplied by itself three times gives cube of that number, e.g.
1 * 1 * 1 = 1
2 * 2 * 2 = 8
3 * 3 * 3 = 27
4 * 4 * 4 = 64
The numbers 1, 8, 27, 64, … are called cube numbers or perfect cubes.

Properties of Perfect Cubes
Property I: If the digit in the one’s place of a number is 0, 1, 4, 5, 6 or 9, then the digit in the one’s place of its cube will also be the same digit.
Property II: If the digit in the one’s place of a number is 2, the digit in the one’s place of its cube is 8, and vice-versa.
Property III: If the digit in the one’s place of a number is 3, the digit in the one’s place of its cube is 7 and vice-versa.
Property IV: Cubes of even natural numbers are even.
Examples: 

Number
Cube
Number
Cube

8

36

512

46656

12

18

1728

5832

Property V: Cubes of odd natural numbers are odd.
Examples:

Number
Cube
Number
Cube

9

35

729

42875

13

17

2197

4913

Property VI: Cubes of negative integers are negative.
Examples:

Number
Cube
Number
Cube

–51

–14

–132651

–2744

–11

–22

–1331

–10648

Patterns in Cubes
I . Adding consecutive odd numbers:

Number
Cube
Sum of consecutive odd numbers
1
13 = 1
1
2
2= 8
3 + 5
3
33 = 27
7 + 9 + 11
4
43 = 64
 13 + 15 + 17 + 19
5
53 = 125
21 + 23 + 25 + 27 + 29

Note that we start with [n * (n – 1) + 1] odd number.
II. Difference of two consecutive cubes:
23 – 13 = 1 + 2 * 1 * 3
33 – 23 = 1 + 3 * 2 * 3
43 – 33 = 1 + 4 * 3 * 3
53 – 43 = 1 + 5 * 4 * 3

III. Cubes and their prime factor:
[Each prime factor of the number appear three times in its cube.]

Number

Prime factorisation

Cube

Prime factors of the cube in its prime factorisations

4

2 * 2


43 = 64


2 * 2 * 2 * 2 * 2 * 2

= 23 * 23


12

2 *  2 * 3


123 = 1728


2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 3

= 23 * 23 * 33


15

3 * 5


153 = 3375


3 * 3 * 3 * 5 * 5 * 5

= 33 * 53


18

2 * 3 * 3


183 = 5832


2* 2 * 2 * 3 * 3 * 3 * 3 * 3 * 3

= 23 * 33 * 33


21

3 * 7


213 = 9261


3 * 3 * 3 * 7 * 7 * 7 = 33 * 73



Example 1: Is 500 a perfect cube?
Solution: 500 = 5 * 5 * 5 * 2 * 2
∵ In the above prime factorisation 2 * 2 remain after grouping the prime factors in triples.
∴ 500 is not a perfect cube.

Points to Remember- Cubes and Cube Roots Class 8 Notes | EduRev

Example 2: Is 1372 a perfect cube? If not, find the smallest natural number by which 1372 must be multiplied so that the product is a perfect cube.
Solution: We have 1372 = 2 * 2 * 7 * 7 * 7
Since, the prime factor 2 does not appear in a group of triples.
∴ 1372 is not a perfect cube.
Obviously, to make it a perfect cube we need one more 2 as its factor.
i.e. [1372] * 2 = [2 * 2 * 7 * 7 * 7] * 2
or
2744 = 2 * 2 * 2 * 7 * 7 * 7
which is a perfect cube.
Thus, the required smallest number = 2.

Points to Remember- Cubes and Cube Roots Class 8 Notes | EduRev

Example 3: Is 31944 a perfect cube? If not then by which smallest natural number should 31944 be divided so that the quotient is a perfect cube?
Solution: We have 31944 = 2 * 2 * 2 * 3 * 11 * 11 * 11
Since, the prime factors of 31944 do not appear in triples as 3 is left over.
∴ 31944 is not a perfect cube. Obviously, 31944 / 3 will be a perfect cube
i.e. [31944] ÷3 = [2 * 2 * 2 * 3 * 11 * 11 * 11] /3
or
10648 = 2 * 2 * 2 * 11 * 11 * 11
∴ 10648 is a perfect cube.
Thus, the required least number = 3.

Points to Remember- Cubes and Cube Roots Class 8 Notes | EduRev

Examples:

Numbers
Number ending in
Perfect cube
One’s digit of the cube

31

22

33

24

35

36

27

58

29

10

1

2

3

4

5

6

7

8

9

0

29791

10648

35937

13824

42875

46656

19683

195112

24389

1000

1

8

7

4

5

6

3

2

9

0

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