Cubes
You have learned that if x is a nonzero number then x* x* x* = x^{3} and is read as the 'cube of x' or simply 'x cubed'.
Therefore 27 = 3 * 3* 3 = 3^{3} or 27 is '3 cubed’.
8 = 2* 2* 2 = 23 or '2 cubed'
Consider the following table


1 
1^{3} = 1* 1* 1 = 1 
2 
2^{3} = 2* 2* 2 = 8 
3 
3^{3} = 3* 3* 3 = 27 
4 
4^{3} = 4 * 4 * 4 = 64 
5 
5^{3} = 5 * 5 * 5 = 125 
6 
6^{3}= 6 * 6 * 6 = 216 
7 
7^{3} = 7 * 7 * 7 = 343 
8 
8^{3} = 8 * 8 * 8 = 512 
9 
9^{3} = 9 * 9 * 9 = 729 
10 
10^{3} = 10 * 10 * 10 = 1000 
11 
11^{3} = 11 * 11 * 11 = 1331 
12 
12^{3} = 12 * 12 * 12 = 1728 
13 
13^{3}= 13 * 13 * 13 = 2197 
14 
14^{3} = 14 * 14 * 14 = 2744 
15 
15^{3} = 15 * 15 * 15 = 3375 
16 
16^{3} = 16 * 16 * 16 = 4096 
17 
17^{3} = 17 * 17 * 17 = 4913 
18 
18^{3} = 18 * 18* 18 = 5832 
19 
19^{3} = 19* 19* 19 = 6859 
20 
20^{3} = 20* 20* 20 = 8000 
These integers 1,8,27 . . . 8000 are called perfect cubes.
A nonzero number x is a perfect cube if there is an integer m such that x = m* m* m
Perfect Cubes
How do we find out whether a given number is a perfect cube?
If a prime p divides a number m then p^{3} will divide m^{3}.
In the prime factorization of a perfect cube, every prime occurs 3 times of a multiple of three times.
For example
In order to check whether a number is a perfect cube or not, we find its prime factors and group together triplets of the prime factors. If no factor is left out then the number is a perfect cube. However if one of the prime factors is a single factor or a double factor then the number is not a perfect cube.
Example : 15
Examine if (i) 392 and (ii) 106480 are perfect cubes.
Solution:
(i)
392=2*2*2*7*7
7 is a double factor, it is not a part of a triplet so it is not a perfect cube.
(ii)
106480
One prime factor 2 and the prime factor 5 are not parts of a triplet so 106480 is not a perfect cube.
Example : 16
Is 19683 a perfect cube?
Solution:
19683
Since the prime factor 3 forms three triplets so 19683 is a perfect cube.
Properties of Cubes
If you look at the table in the first page, you will notice that numbers with their units digits 1, 4, 5, 6 or 9 have perfect cubes whose units digits are also 1, 4, 5, 6, 9 respectively.
A number with a units digit of 2 has a cube whose units digit is 8 and viceversa.
A number with a units digit of 3 has a cube whose units digit is 7 and viceversa.
The cube of a negative number is negative.
(1)^{3} = 1 1 1 = 1 = 1^{3}
(2)^{3} = 2 2 2 = 8 =  (2)^{3} etc.
So, negative numbers can be cubes.
Finding the Cubes of a Number
We can find the cube of a number by multiplying the number with itself three times. Another method which is based on the algebraic identity can also be used. Recall that to find the square of a two digit number you used (a + b)^{2} = a^{2} + 2ab + b^{2} and you formed three columns and worked out the square.
Here we will use the algebraic identity (a + b)^{3} = a^{3} + 3a^{2}b +3ab^{2} +b^{3} and form 4 columns and we will find the cube by a similar method as that of the squares.
Example : 17
Find the cube of 89, using the alternate method.
Solution:
We take a = 8 b = 9.
I
a^{3} 
II
3a^{2}b 
III
3ab^{2} 
IV
b^{3} 
8^{3} 
3 * 8^{2} * 9 
3 * 8 * 9^{2} 
9^{3} 
512 
3 * 64 * 9 
24 * 81 
729 
512 
1728 
1944 
729 
+ 192 
+ 201 
+ 72 

704

1929

2016


(89)^{3} = 704969
Reasoning:
Make 4 columns. In the first column write a^{3}, second 3a^{2}b, third 3ab^{2}, and in the fourth write b^{3}.
Write the values of each.
In the first we get a^{3} = 8^{3} = 512
In the second, we get 3a2b = 3* 8^{2}* 9 = 1728
In the third, we get 3ab2 = 3 * 8 * 9^{2} = 1944
In the fourth, we get b^{3} = 9^{3} = 729
Underline the digit of b^{3} in this case 9. Add the digits 72 to the value of 3ab^{2} in this case 1944
Underline the units digit, in this case 6. Add the digits 201 to the value of 3a^{2}b in this case 1728
we get
Underline the units digit, in this case 9. Add the digits 192 to the value of a3 in this case 512
we get
Underline all the digits. The required cube is 704969. We combine all the underlined digits to get the value of the number required.
Example :18
Examine if 53240 is a perfect cube. If not, find the smallest number by which it must be multiplied to form a perfect cube. Also find the smallest number by which it must be divided to form a perfect cube.
Solution:
Reason
Find the prime factors of 53240
53240
5 is not a part of a triplet.
For 53240 to become a perfect cube we need to multiply it by 5 * 5 = 25.
The smallest number by which 53240 must be multiplied to form a perfect cube is 25.
If we divide 53240 by 5 then the resulting number will be a perfect cube. So 5 is the least number by which 53240 must be divided to obtain a perfect cube.
Cube Roots
If n = m^{3} the m is the cube root of n. We write this as m = or n^{1/3}
From Table 2 we have
1^{3} = 1 
so 

2^{3} = 8 
so 

3^{3} = 27 


4^{3} = 64 


5^{3} = 125 


6^{3} = 216 


7^{3} = 343 


8^{3} = 512 


9^{3} = 729 


10^{3} = 1000 


Cube Root by Prime Factorization Method
You have already seen that in the prime factorization of a perfect cube, primes occur as triplets. We therefore can find using the following algorithm.
Step 1 Find the prime factorization of n.
Step 2 Group the factors in triplets such that all three factors in triplet are the same.
Step 3 If some prime factors are left ungrouped, the number n is not a perfect cube and the process stops.
Step 4 If no factor is left ungrouped, choose one factor from each group and take their product. The product is the cube root of n.
Example 19
Find the cube root of a) 91125 b) 551368
Solution:
a)
Reason
 Find the prime factorization of 91125
 Group the prime factors as triplets such that all the factors in each triplet are the same
 Choose a factor from each triplet
 Multiply and get your answer
91125
b)
Reason
Find the prime factors of 551368 Group the factors in triplets each factor of the triplet being the same.
Select a factor from each triplet multiply these factors. Write your answer.
Cube Root using Units Digits
Perfect cubes which are six digit numbers can be obtained using the method of the units digits.
If a six digit perfect cube has a units digit of 0, 1, 4, 5, 6 or 9 then its cube root will have a units digit of 0, 1, 4, 5, 6 or 9.
If however the units digit of the cube is 8 then the units digit of the cube root will be 2.
If the units digit of the cube is 2 then the units digit of the cube root will be 8.
If the units digit of the cube is 7, the units digit of the cube root will be 3 and if the units digit of the cube is 3 the units digit of the cube root will be 7.
The cube root of a six digit perfect cube will have at the most, two digits, because the least seven digit number 1000000 = 1003 and its cube root 100 is a three digit number.
We determine the two digits of the cube root as follows:
Step 1 Look at the digit in the units place of the perfect cube and determine the digit in the units place of the cube root as discussed above.
Step 2 Strike out from the right, the last three digits, that is, the units the tens and the hundreds digits. If nothing is left we stop. The digit in step 1 is the cube root.
Step 3 Consider the digits left over from Step 2. Find the largest single digit number whose cube is less than or equal to those left over digits. This is the tens digit.
Example : 20
Find the cube roots of the following numbers
a) 512 b) 2197 c) 117649
Solution:
 512
The digit in the units place is 2. Therefore the digit in the units place of its cube root is 8.
Strike out the 3 digits – the units, the tens and the hundreds.
No number is left.
The required cube root is 8.
or
 2197
Units digit of the cube root is 3
2
1^{3} = 1 < 2
1 = tens digit
=13
Reason
Units digit of 2197 is 7 so the units digit of its cube root is 3.
Strike out the units, tens and hundreds digitsDigit left is 2.
Find the largest single digit number whose cube is less than or equal to this left over digit 2 In this case it is 1.
 117649
Units digit of the cube root = 9
117
4^{3} = 64 < 117 < 125 = 53
4 is the tens digit
Reason
Units digit of 117649 is 9. So the units digit of its cube root is 3.
Strike out the units, tens and hundreds digits number left.
The largest single digit number whose cube 64 is less than 117 is 4.
Try these problems
 Find the cubes of
 402
 819
 Using the alternative method to find the cubes of
 56
 87
 Find the smallest number by which the following numbers must be multiplied to obtain a perfect cube. Also find the cube root of the resulting number.
 137592
 107811
 35721
 Find the smallest number by which the following numbers must be divided so that the products are perfect cubes. Also find the cube root of the resulting number.
 7803
 8192
 26244
 Fill in the blanks by observing the pattern.
9* 12 + 13 = 10 = 12 * 10
8 * 22 + 23 = 40 = 22 * 10
–* – + – = 90 = 32 * 10
6* 42 + 43 = 160 = ––* ––
5* 52 + 53 = –– = ––* ––
 Find the cube roots of the following numbers by finding their units and tens digits.
 389017
 91125
 46656
 110592
Answers to Practice Problems
 a. Cube of 402 = (402)^{3}
= 402 * 402 * 402
= 64964808
b. Cube of 819 = (819)^{3}
= 819 * 819 * 819
= 549353259
 a. Consider 56 a = 5 and b = 6
I
a^{3} 
II
3a^{2}b 
III
3ab^{2} 
IV
b^{3} 
Check examples for reasons. 
53
125
+50
175 
3 * 52 * 6
450
+56
506 
3 * 5 * 62
540
+21
561 
63
216 
56^{3} = 175616

b. Consider 87 a = 8 and b = 7
I
a^{3} 
II
3a^{2}b 
III
3ab^{2} 
IV
b^{3} 
Check examples for reasons. 
83
512
+146
658 
3 * 82 * 7
1344
+121
1465 
3 * 8 * 72
1176
+34
1210 
73
343 
87^{3} = 658503 
 a.
137592 =
In order to obtain a perfect cube we need 7 to complete the triplet and 13 * 13 to complete the next unfinished triplet.
The least number by which 137592 must be multiplied to form a perfect cube is
7 * 13 * 13 = 1183.
b.
107811 =
The extra factor 3 is not a part of a triplet. So we need 3 * 3, to get a perfect cube.
The least number by which 107811 must be multiplied is 3 * 3 = 9 so that we get a perfect cube.
c.
We need a 7 to complete the triplet.
The least number by which 35721 must be multiplied to form a perfect triplet is 7.

a.
17 *17 is the least factor needed to divide the number 7803 to get a perfect cube.
289 is the least by which 7803 must be divided to obtain a perfect square.
b.
2 is the least factor which is not a part of a triplet.
2 is the least number by which 8192 must be divided so that it forms a perfect cube.
c.
2 * 2 and 3 * 3 are not parts of triplets
2 * 2 * 3 * 3 = 36 is the least number by which 26244 must be divided to obtain a perfect cube.

9 * 12 + 13 = 10 = 12 *10
9 * 12 + 13 = 10 = 12 *10
8 * 22 + 23 = 40 = 22*10
7 * 32 + 33 = 90 = 32*10
6 * 42 + 43 = 160 = 42*10
5 * 52 + 53 = 250 = 52 *10

a. 
389017
Units digit of the cube root = 3
Number left = 389
7^{3} = 343 < 389 < 512 = 83
Tens digit of the cube root is 7.

Reason
Units digit of 389017 is 7
Units digit of the cube root is 3.
Strike out the units, tens, and hundreds digits.
The largest single digit number whose cube 343 is less than 389 is 7. 
b. 
91125
Units digit of the cube root = 5
Number left = 91
4^{3} = 5 < 91 < 125 = 53
Tens digit of the cube root is 4.

Reason
Units digit of 91125 is 5.
Units digit of the cube root is 5.
Strike out the units, tens, and hundreds digits.
The largest single digit number whose cube 64 is less than 91 is 4. 
c. 
46656
Units digit of the cube root = 6
Number left = 46
3^{3} = 27 < 46 < 64 = 43
Tens digit of the cube root = 3

Reason
Units digit of 46656 is 6.
Units digit of the cube root is 6.
Strike out the units, tens, and hundreds digit.
The largest single digit number whose cube 27 is less than 46 is 3. 
d. 
110592
Units digit of the cube root = 8
Number left = 110
4^{3} = 64 < 110 < 125 = 53
Tens digit of the cube root = 4

Reason
Units digit of 110592 is 2
The cube root's units digit is 8.
Strike off the units, tens, and hundreds digits.
The single largest digit whose cube 64 is less than 110 is 4. 