Chapter 8 - Cubes and Cube Roots
Cube : Cube of X is X x X x X and is denoted as .
Cube Root : Cube root of is X and is denoted as = X . means .
Note : Memorise the cubes of the following numbers;
= 8 ; = 27 ; = 64 ; = 125 ; = 216 ; = 343 ; = 512 ; = 729
= 1000 ; = 1331 ; = 1728 ; = 2197 ; = 3375 ; = 15625
Finding Cube of a number :
We know = + + + .
Cube of a number consists of four parts / / / .
Here all the parts except the first part ( ) will have (No. of digits in the number - 1) digits.
Example 1 : Find .
= / 3 x x 5 / 3 x 1 x / = 1 / 15 / 75 / 125.
As we did with the squares of numbers, now we have to start from our right hand side with
(2 - 1) i.e. 1 digit in each part. So the 12 in the fourth part is carried over to the third part - 75 and added. So the third part becomes 87. Now the extra 8 is to be carried over to the next part
- 15 and added and so the second part becomes 23 and the extra 2 is carried over to the first part and so the first part becomes 3.
Hence the required answer is 3 3 7 5.
Example 2 : Find the value of .
= / 3 x x 8 / 3 x 2 x / = 8 / 96 / 384 / 512
51 => 435
8 / 96 / 5 / 2
43 = > 139
13 => 21 = 21952.
Example 3 : Find the value of .
= / 3 x x 3 / 3 x 3 x / = 27 / 81 / 81 / 27
2 => 83
27 / 81 / 3 / 7
8 => 89
8 => 35 = 35937.
Example 4 : Find the value of .
= / 3 x x 5 / 3 x 4 x / = 64 / 240 / 300 / 125
12 => 312
64 / 240 / 312 / 5
31 => 271
64 / 1 / 2 / 5
27 = 91 = 91125.
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Example 12 : = / 3 x x 7 / 3 x 1 x / = 1 / 21 / 147 / 343.
Here each part is to have 3 digits but the second part has only 2 digits and so the second part
21 is to be prefixed with a ‘0’ to make it a 3 digit number and the required answer = 1021147343.
Example 13 : = 216 / 540 / 450 / 125 = 216540450125.
Note : The difference between the cubes of two consecutive numbers =
(3 x first no. x second no.) + 1. For example the difference between and = 3 x 10 x
11 + 1 = 331. Similarly the difference between and = 3 x 19 x 20 + 1 = 1141.
Shortcut Method for finding Cube Roots of perfect cubes :
Since the conventional method is time consuming we will be using a Shortcut Method.
= 1. So if the last digit of the cube is 1, the last digit of the cube root is 1.
= 8. So if the last digit of the cube is 8, the last digit of the cube root is 2.
= 27. So if the last digit of the cube is 7, the last digit of the cube root is 3.
= 64. So if the last digit of the cube is 4, the last digit of the cube root is 4.
= 125. So if the last digit of the cube is 5, the last digit of the cube root is 5.
= 216. So if the last digit of the cube is 6, the last digit of the cube root is 6.
= 343. So if the last digit of the cube is 3, the last digit of the cube root is 7.
= 512. So if the last digit of the cube is 2, the last digit of the cube root is 8.
= 729. So if the last digit of the cube is 9, the last digit of the cube root is 9.
= 1000. So if the last digit of the cube is 0, the last digit of the cube root is 0.
So we come to know that in respect of 1, 4, 5, 6, 9 and 0 , the respective numbers are repeated.
In respect of 2, 3, 7 and 8, the respective complement of 10 will come.
Step 1 : Group the number from RHS (Right Hand Side) with 3 digits in a group.
Step 2 : From the last digit of the group on your RHS, we can arrive the last digit (i.e. the digit at the units place) of the cube root.
Step 3 : Take the next group and take the immediate lower perfect cube of the said number.
The cube root of the same will be the digit at the tenth place.
Example 1 : Find the cube root of 4913. Step 1 : 4,913.
Step 2 : In 913, the last digit is 3 and when the last digit of the perfect cube is 3, the cube root can end with 10’s complement of 3 i.e. 7. So the digit at the units place is 7.
Step 3 : The immediate lower perfect cube of 4 is 1 and the cube root of the same is 1.
Therefore the required answer is 17.
Example 2 : Find the cube root of 85184. Step 1 : 85,184.
Step 2 : In 184, the last digit is 4 and hence 4 itself is the last digit of the cube root.
Step 3 : The immediate lower perfect cube of 85 is 64 and the cube root of the same is 4.
Therefore the required answer is 44.
Example 3 : Find the cube root of 614125. Step 1 : 614,125.
Step 2 : In 125, the last digit is 5 and hence 5 itself is the last digit of the cube root.
Step 3 : The immediate lower perfect cube of 614 is 512 and the cube root of the same is 8.
Therefore the required answer is 85.
Example 4 : 238328 => 238,328. The last digit is (10 - 8) = 2 and the first digit is the cube root of 216 i.e. 6. Therefore the required answer is 62.
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22.
If X - = 10, find the value of - .
Solution :
We know if X - = a, then - = + 3a . So here it is 1000 + 3 x 10 = 1030.
23.
If X - = 6, then find the value of + ;
Shortcut Method : If X - = a, then + = + 2. So here it is 36 + 2 = 38.
Similarly if X + = a, then + = - 2.
24.
If X + = 5, find the value of + .
Solution :
If X + = 5. Squaring LHS and RHS we get + + 2 = 25.
i.e. + = 25 - 2 = 23. Again squaring LHS and RHS we get + + 2 = 529.
So we get + = 529 - 2 = 527.
- 2.
Shortcut Method : If X + = a, then the value of + =
Similarly if X - = a, then the value of + =
25.
If X + = 6, then + is.
Solution :
It is given = 6. Therefore we get =
- 2.
. i.e. we get = 36.
So we get = 34.
So we get = = 39304.
1
i.e. = 39304. i.e.
X X
1
2 x
1
X 4
+ = 39304 - 3 x 34 = 39202.
Shortcut Method : If X + = a , then the value of =
Similarly if = a, then the value of = .
Note : The above problems are frequently found in tests especially Staff Selection Commission
Tests. If the direct formulas as above are memorised we can save a significant amount of time.
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