Cubes and Cube Roots - Class 8 : Notes
Cube:
It is a three-dimensional figure made of six equal square sides.
Cube number or Perfect cubes:
It is a number which is the product of three same numbers.
Example: Cube number of 2 will be 2 x 2 x 2 = 8. Thus, 8 is a cube number.Â
Cubes of some natural numbers:
Number |
Cube |
Number |
Cube |
1 |
1 x 1 x 1 = 1 |
11 |
11 x 11 x 11 = 1331 |
2 | 2 x 2 x 2 = 8 | 12 |
12 x 12 x 12 = 1728 |
3 |
3 x 3 x 3 = 27 |
13 |
13 x 13 x 13 = 2197 |
4 | 4 x 4 x 4 = 64 | 14 |
14 x 14 x 14 = 2744 |
5 |
5 x 5 x 5 = 125 |
15 |
15 x 15 x 15 = 3375 |
6 | 6 x 6 x 6 = 216 | 16 |
16 x 16 x 16 = 4096 |
7 |
7 x 7 x 7 = 343 |
17 |
17 x 17 x 17 = 4913 |
8 | 8 x 8 x 8 = 512 | 18 |
18 x 18 x 18 = 5832 |
9 |
9 x 9 x 9 = 729 |
19 |
19 x 19 x 19 = 6859 |
10 | 10 x 10 x 10 = 1000 | 20 |
20 x 20 x 20 = 8000 |
... |
... |
... |
... |
Properties of Cube Numbers:
1. The cube of an even number will always be an even number.
Example: 8^{3} = 512, 12^{3} = 1728, etc.Â
2. The cube of odd number will always be an odd number.
Example: 7^{3} = 343, 19^{3} = 6589, etc.Â
3. If the cube number have x at its oneâ€™s digit or unitâ€™s place then it always end with the digit as shown in the table below:
Unitâ€™s digit of number |
Last digit of its cube number | Example |
1 |
1 |
11^{3} = 1331, 21^{3} = 9261, etc. |
2 | 8 |
2^{3} = 8, 12^{3} = 1728, 32^{3} = 32768, etc. |
3 |
7 |
13^{3} = 2197, 53^{3} = 148877, etc. |
4 | 4 |
24^{3} = 13824, 74^{3} = 405224, etc. |
5 |
5 |
15^{3} = 3375, 25^{3} = 15625, etc. |
6 | 6 |
6^{3} = 216, 26^{3} = 17576,etc. |
7 |
3 |
17^{3} = 4913, 37^{3} = 50653,etc. |
8 | 2 |
8^{3} = 512, 18^{3} = 5832, etc. |
9 |
9 |
19^{3} = 6859, 39^{3} = 59319, etc. |
10 | 20 |
10^{3} = 1000, 20^{3} = 8000, etc. |
Example 1: Find the oneâ€™s digit for 27.
Solution:Â As the last digit of given number is 7, So the oneâ€™s digit for 27â€™s cube number will be 3.
Example 2: Find the oneâ€™s digit for 149.
Solution:Â As the last digit of given number is 9, So the oneâ€™s digit for 149â€™s cube number will be 9.
Interesting patterns of Cube Number:
1. Addition of consecutive odd numbers will give Cube Number-
1^{3}Â Â Â Â Â Â Â = Â Â Â Â Â Â Â Â 1 Â Â Â Â Â Â Â Â =Â Â Â Â Â Â Â Â Â 1
2^{3}Â Â Â Â Â Â Â = Â Â Â Â Â Â Â Â 8 Â Â Â Â Â Â Â Â = Â Â Â Â Â Â Â Â 3 + 5
3^{3}Â Â Â Â Â Â Â = Â Â Â Â Â Â Â Â 27Â Â Â Â Â Â Â = Â Â Â Â Â Â Â Â 7 + 9 + 11
4^{3}Â Â Â Â Â Â Â = Â Â Â Â Â Â Â Â 64 Â Â Â Â Â Â = Â Â Â Â Â Â Â Â 13 + 15 + 17 + 19
5^{3}Â Â Â Â Â Â Â = Â Â Â Â Â Â Â Â 125Â Â Â Â Â = Â Â Â Â Â Â Â Â 21 + 23 + 25 + 27 + 29
2. Cubes and their prime factors-
The prime factors of any cube number will be in pair of 3.
Example:
(i) 4^{3} = 64 = 2 x 2 x 2 x 2 x 2 x 2 = 2^{3} x 2^{3
}(ii)12^{3} = 1728 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 = 2^{3} x 2^{3} x 3^{3
}
Note:
(i) If any prime factor is not in pair of 3, then it will not be a perfect cube.
(ii)Â Numbers like 1729, 4104, 13832, are known as Hardy â€“ Ramanujan Numbers. They can be expressed as sum of two cubes in two different ways.
Example 1: Is 128 a perfect cube number?
Solution:Â After finding prime factors of 128, we can write 128 = 2 x 2 x 2 x 2 x 2 x 2 x 2
We can see that each prime factor is not in pair of 3. Hence, 128 is not a perfect cube number.
Example 2: Find the smallest number by which 675 must be multiplied to obtain a perfect cube.
Solution:Â On finding prime factors of 675, we have 675 = 3 x 3 x 3 x 5 x 5.
We can see that, triplets of 5 is missing. Hence, on multiplying given number by 5 we can have a perfect cube number.Â Thus, 675 x 5 = 3375 which is a perfect cube number.
Example 3: Find the smallest number by which 192 must be divided to obtain a perfect cube.
Solution:Â On finding prime factors of 192, we have 192 = 2 x 2 x 2 x 2 x 2 x 2 x 3.
We can see that, triplets of 3 is missing while other numbers have triplets. Hence, on dividing given number by 3 we can have a perfect cube number.
Thus, 192 / 3 = 64 which is a perfect cube number.
Cube Roots:
It is the inverse operation of finding a cube.Â Symbol âˆ› represents a cube-root.
Example: âˆ›8 = 2, âˆ›216 = 6, etc.
Methods to find a cube root:
1. Prime factorisation method:
Follow the steps given below to understand this method:
Step 1: Find all the prime factors of given cube number.
Step 2: Make as many group of 3 for all common digit.
Step 3: Replace group of 3 by respective single digit.
Step 4: Product of these single digits will give the cube root.
Example 1: Find cube root of 3375.
Solution:
Step 1: Find all the prime factors of given cube number.
The prime factors of 3375 will be 5, 5, 3, 3, 3, 5.
Step 2: Making group of 3 for every common digit.
Here, we get (3 x 3 x 3) and (5 x 5 x 5)
Step 3: Replacing group of 3 by respective digit.
Thus, we get 3 and 5.
Step 4: Taking product of digits.
We get, 3 x 5 = 15.
Thus, âˆ›3375 = 15.
Example 2: Find cube root of 46656.
Solution:Â The prime factors of 46656 are 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3.
On grouping, we have 46656 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 3 x 3 x 3.
So, âˆ›46656 = 2 x 2 x 3 x 3 = 36.
2. Estimation Method:
Follow the steps given below to understand this method:
Step 1: For given cube number start making groups of three digits starting from the right most digit of the number. We can estimate the cube root of a given cube number through a step by step process.
Step 2: First group will give you the oneâ€™s (or unitâ€™s) digit of the required cube root.
Step 3: Now take another group, let it be xyz. Find, a^{3} < xyz < b^{3}. We take the oneâ€™s place, of the smaller number a^{3} as the tenâ€™s place of the required cube root.
Step 4: The digits obtained in step 2 and step 3 will give the final result.
Example 1: Find the cube root of 17576.
Solution:
Step 1: We will form groups of three starting from the rightmost side of 17567.
Thus, the two groups formed will be 576 and 17.
Step 2: For the group 576, the number at unitâ€™s place will be 6.
Step 3: The other group is 17. The group 17 lies between 2^{3} and 3^{3}. Now, as per the method we will take the smaller number which is 2 in this case.
Step 4: Combining the two digits obtained in step 2 and 3, we get 26.
Thus, âˆ›17576 = 26.
Example 2: Guess the cube root of 4913.
Solution:The two groups of 4913 will be 4 and 913.
For group 913, as the last digit is 3 so its cube root will have 7 at its unit place.
For group 4, it will lie between 1^{3} < 4 < 2^{3}. Taking the smaller number which is 1 in this case.
Thus, âˆ›4913 = 17.
Very good
Helped a lot for a overview
But,hardy ramanujan numbers should me expalined more with proper examples
Apart from this it was very nice