Cube and Cube Roots

Cubes

You have learned that if x is a non-zero number then x* x* x* = x³ and is read as the 'cube of x' or simply 'x cubed'.

Therefore 27 = 3 * 3* 3 = 3³ or 27 is '3 cubed’.

8 = 2* 2* 2 = 23 or '2 cubed'

Consider the following table

Digit	Cube of the digit
1	1³ = 1* 1* 1 = 1
2	2³ = 2* 2* 2 = 8
3	3³ = 3* 3* 3 = 27
4	4³ = 4 * 4 * 4 = 64
5	5³ = 5 * 5 * 5 = 125
6	6³= 6 * 6 * 6 = 216
7	7³ = 7 * 7 * 7 = 343
8	8³ = 8 * 8 * 8 = 512
9	9³ = 9 * 9 * 9 = 729
10	10³ = 10 * 10 * 10 = 1000
11	11³ = 11 * 11 * 11 = 1331
12	12³ = 12 * 12 * 12 = 1728
13	13³= 13 * 13 * 13 = 2197
14	14³ = 14 * 14 * 14 = 2744
15	15³ = 15 * 15 * 15 = 3375
16	16³ = 16 * 16 * 16 = 4096
17	17³ = 17 * 17 * 17 = 4913
18	18³ = 18 * 18* 18 = 5832
19	19³ = 19* 19* 19 = 6859
20	20³ = 20* 20* 20 = 8000

These integers 1,8,27 . . . 8000 are called perfect cubes.

A non-zero 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 p³ will divide m³.

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=22277 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 vice-versa. A number with a units digit of 3 has a cube whose units digit is 7 and vice-versa.
The cube of a negative number is negative. (-1)³ = -1 -1 -1 = -1 = -1³ (-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 = a² + 2ab + b² and you formed three columns and worked out the square. Here we will use the algebraic identity (a + b)³ = a³ + 3a²b +3ab² +b³ 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 a³	II 3a²b	III 3ab²	IV b³
8³	3 * 8² * 9	3 * 8 * 9²	9³
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 a³, second 3a²b, third 3ab², and in the fourth write b³.

Write the values of each.

In the first we get a³ = 8³ = 512
In the second, we get 3a2b = 3* 8²* 9 = 1728
In the third, we get 3ab2 = 3 * 8 * 9² = 1944
In the fourth, we get b³ = 9³ = 729

Underline the digit of b³ in this case 9. Add the digits 72 to the value of 3ab² in this case 1944

Underline the units digit, in this case 6. Add the digits 201 to the value of 3a²b 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 = m³ the m is the cube root of n. We write this as m =

or n^1/3

From Table 2 we have

1³ = 1	so
2³ = 8	so
3³ = 27
4³ = 64
5³ = 125
6³ = 216
7³ = 343
8³ = 512
9³ = 729
10³ = 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:

a)	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

b)	2197
	Units digit of the cube root is 3 2 1³ = 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.
c)	117649
	Units digit of the cube root = 9 117 4³ = 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

Using the alternative method to find the cubes of

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

Consider 56 a = 5 and b = 6

I a³	II 3a²b	III 3ab²	IV b³	Check examples for reasons.
53 125 +50 175	3 * 52 * 6 450 +56 506	3 * 5 * 62 540 +21 561	63 216	Check examples for reasons.
56³ = 175616

Consider 87 a = 8 and b = 7

I a³	II 3a²b	III 3ab²	IV b³	Check examples for reasons.
83 512 +146 658	3 * 82 * 7 1344 +121 1465	3 * 8 * 72 1176 +34 1210	73 343	Check examples for reasons.
87³ = 658503

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.

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.

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 7³ = 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 4³ = 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 3³ = 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 4³ = 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.