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CHAPTER
2
CUBES AND CUBE ROOTS
2.1 Introduction
Like squares and square roots in Chapter 1, we shall study cubes and cube roots in this
Chapter. We first discuss some properties of perfect cubes. We will then discuss patterns
of perfect cubes which help us in finding cube roots of small perfect cube numbers. Based
on the relationship between units digits of numbers and units digits of their cubes, we
discuss a method of finding the cube root of a perfect cube. This method applies to numbers
having at the most six digits. The advantage of this method is that there are very few and
simple calculations. We then discuss prime factorisation method of finding the cube roots.
Like square roots, we do have a division method to find the cube root of a number.
However, we do not discuss this method here, as it is a little difficult and beyond the scope
of this book.
2.2 Cube of a Number and Perfect Cube Numbers
We know that if x is a non-zero number, then x × x × x, written as x 3, is called the
cube of x or simply x cubed. Thus, 8 (= 2 × 2 × 2) is the cube of 2 or 2 cubed. Similarly,
27 (= 3 × 3 × 3) is the cube of 3 or 3 cubed. Table 2.1 gives the cubes of the single
digits 1 to 9.
Table 2.1 : Cubes of the digits 1 to 9
x
1
2
3
4
5
6
7
8
9
x3
1
8
27
64
125
216
343
512
729
Each of the numbers 1, 8, 27, . . ., 729 is the cube of some integer or the other. Such
numbers are called perfect cubes (or perfect third powers ).
A number n is a perfect cube if there is an integer m such that n=m × m × m.
CUBES AND CUBE ROOTS
29
Perfect cubes grow very fast. As m grows from 1 to 9, the perfect cube m3 grows
from 1 to 729. Perfect cubes are fairly scattered. Upto 100, there are only four perfect
cubes. Upto 1000, there are only ten perfect cubes. (Are you trying to check this fact?
Note that 103 = 1000.)
How do we examine whether a given number is a perfect cube?
If a prime p divides m, then p× p× p will divide m × m × m, i.e., m3. Therefore,
if a prime p divides a perfect cube, then p3 also divides this perfect cube.
In other words, in the prime factorisation of a perfect cube, every prime occurs three
times or a multiple of three times. For example,
64 = 2 × 2 × 2 × 2 × 2 × 2
(= 26)
27000 = 2 × 2 × 2 × 3 × 3 × 3 × 5 × 5 × 5 ( = 23 × 33 × 53)
Thus, to check whether a number is a perfect cube, we find its prime factorisation and
group together triples of the same prime factors. If no factor is left out, the number is
a perfect cube. However, if some factor is left as a single factor or a double factor, then the
number is not a perfect cube.
Example 1 : Examine if (i) 392 and (ii) 106480 are perfect cubes.
Solution : (i) 392 = 2 × 2 × 2 × 7 × 7
Here, 7 does not appear in a group of three. Hence, 392 is not a perfect cube.
(ii) 106480 = 2 × 2 × 2 × 2 × 5 × 11 × 11 × 11
Here, one prime factor 2 and one prime factor 5 do not appear in a group of three.
Therefore, 106480 is not a perfect cube.
Example 2 : Examine if 53240 is a perfect cube. If not, find the smallest number by
which it must be multiplied so that the product is a perfect cube. Find also the smallest
number by which it must be divided so that the quotient is a perfect cube.
Solution : 53240 = 2 × 2 × 2 × 5 × 11 × 11 × 11
Here, the prime 5 does not appear in a group of three. Hence, 53240 is not a perfect
cube. Further, 5 appears only once. If we multiply the number by 5 × 5, then in the product,
5 will also appear in a group of three and the product will be a perfect cube. Thus, the
required smallest number by which the given number should be multiplied is 5 × 5, i.e., 25.
Finally, if we divide the given number 53240 by 5, the resulting number has prime
factors in a group of three. In fact, 53240 ÷ 5 = 10648 = 2 × 2 × 2 × 11 × 11 × 11.
Hence, the required number in this case is 5.
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MATHEMATICS
Remarks 1 : From Table 2.1, we observe that cubes of the digits 1, 4, 5, 6 and 9 are
numbers ending in the same digits 1, 4, 5, 6 and 9, respectively. However, 2 and 8 make
a pair in the sense that the cube of 2 ends in 8 and the cube of 8 ends in 2. Numbers 3 and
7 make another pair of this kind, i.e., the cube of either ends in the other (33 = 27, 73 = 343).
Further, 103 = 1000 shows that if a number ends in a zero, its cube will end in three zeros.
These observations will help us in finding the cube roots of perfect cubes.
2. If a number is negative, then its cube is also negative. For example,
(–1)3 = (–1) × (–1) × (–1) = –1 = –13
(–2)3 = (–2) × (–2) × (–2) = – 8 = – 23
(–5)3 = –125 = –53, (– m)3 = – m3
This shows that negative numbers may also be perfect cubes. This is in contrast to perfect
squares, which are never negative.
2.3 Finding the Cube of a Two-digit Positive Number (Alternative Method)
The cube of a number can be obtained by multiplying the number with itself three times. To
find x3, we may first find x2 and then x2 × x. Here we shall discuss an alternative method
of finding x3, when x is a two digit number.
Let x = ab, where a is the tens digit and b is the units digit. Recall that for (ab)2, we
formed three columns a2 | 2a × b | b2. Like in the method of finding x2, here too we form
columns. We use the Identity
(a + b)3 = a3 + 3a2 b + 3ab2 + b3.
(You will learn this Identity in Chapter 6.) For (ab)3 we form four columns
a3 | 3a2 × b | 3a × b2 | b3
Rest of the procedure is as before, i.e., we retain the units digit after addition and
take the remaining digits to the next column for addition.We shall illustrate the method
by means of examples.
Example 3 : Find 423 using alternative method.
Solution : Here, a = 4, b = 2. Thus, the four columns are :
a3 | 3a2 × b | 3a × b2 | b3
4
3
= 64
+ 10
74
74
∴
2
3×4×2 3×4×2
= 96
+4
100
0
423 = 74088
= 48
2
2
3
=8
8
8
CUBES AND CUBE ROOTS
31
Example 4 : Find 873 by alternative method.
Solution : Here, a = 8, b = 7. In a situation where a and b are not small, it may not be
possible to calculate 3a2 × b and 3a × b2 quickly. In such a situation, we simplify the
working as follows :
64
49
64
49
´ 8 ´ 21
´ 24 ´ 7
2
2
2
2
a
b
b
a
512 1344
1176 343
a
3b
3a
b
+146 + 121 + 34
,
i.e.,
3
2
2
3
1210
658 1465
a 3a × b 3a × b b
∴
658
873 = 658503
5
0
Example 5 : Find the cubes of the following numbers using alternative method :
(i) 27
(ii) 45
(iii) 81
Solution : (i) For 273, we have
49
4
4
6
21
2
294
8
84
+11 +32 + 34
328
19 116
8
19
6
∴
49
7
343
3
273 = 19683
(ii) For 453, we have
16
25 25
16
15
12
5
4
64 240 300 125
+27 + 31 + 12
91 271 312
2
5
1
91
∴
453 = 91125
3
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MATHEMATICS
(iii) For 813 , we have
64 64
3
8
512 192
+19 +2
531 194
531
∴
1
24
24
1
1
1
4
1
4
813 = 531441
EXERCISE 2.1
1.
Write the units digit of the cube of each of the following numbers :
31, 109, 388, 833, 4276, 5922, 77774, 44447, 125125125
2.
Find the cubes of the following numbers:
(i) 35
(ii) 56
(iii) 72
(iv) 402
(v) 650
(vi) 819
3.
Find the cubes of the following numbers using alternative method :
(i) 35
(ii) 56
(iii) 72.
4.
Which of the following numbers are not perfect cubes?
(i) 64
(ii) 216
(iii) 243
(iv) 1728
5.
For each of the non-perfect cubes in Question 4, find the smallest number by which it
must be multiplied so that the product is a perfect cube.
6.
For each of the non-perfect cubes in Question 4, find the smallest number by which it
must be divided so that the quotient is a perfect cube.
7.
By taking three different values of n, verify the truth of the following statements :
(i) If n is even, then n3 is also even.
(ii) If n is odd, then n3 is also odd.
(iii) If n leaves remainder 1 when divided by 3, then n3 also leaves 1 as remainder
when divided by 3.
(iv) If a natural number n is of the form 3p + 2, then n3 is also a number of the
same type.
CUBES AND CUBE ROOTS
8.
33
Write true (T) or false (F) for the following statements :
(i) 392 is a perfect cube.
(ii) 8640 is not a perfect cube.
(iii) No perfect cube can end with exactly two zeros.
(iv) There is no perfect cube which ends in 4.
(v) For an integer a, a3 is always greater than a2.
(vi) If a and b are integers such that a2 > b2, then a3 > b3.
(vii) If a divides b, then a3 divides b3.
(viii) If a2 ends in 9, then a3 ends in 7.
(ix) If a2 ends in 5, then a3 ends in 25.
(x)
If a2 ends in an even number of zeros, then a3 ends in an odd number of zeros.
2.4 Cube Roots
If n is a perfect cube, then for some integer m, n = m3. Here, the number m is called the
cube root of n. Thus, a number m is a cube root of a number n, if m3 = n. For example :
2 is a cube root of 8, because 23 = 8.
5 is a cube root of 125, because 53 = 125.
11 is a cube root of 1331, because 113 = 1331.
If m is a cube root of n, we write m =
3
n.
Thus, 2 = 3 8 , 5 = 3 125 and 11 = 3 1331 .
In Tables 2.2 and 2.3, we list respectively all the perfect cubes upto 1000 and their cube roots.
Table 2.2
Table 2.3
m
1
2
3
4
5
6
7
8
9
10
m3
1
8
27
64
125
216
343
512
729
1000
n
1
8
27
64
125
216
343
512
729
1000
3
n
1
2
3
4
5
6
7
8
9
10
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MATHEMATICS
Remark : We use the symbol ‘3 ’ to denote cube root in the manner we
used ‘2 ’ to denote square root. It is a mere convenience that we omit 2 and use ‘
’ to
denote square root. Strictly speaking, we should use the symbol ‘2 ’ for square root.
However, we never omit 3 while writing ‘3 ’ for cube roots.
We now discuss some methods of finding cube roots of perfect cubes.
2.5 Cube Root Through a Pattern
Like squares of natural numbers, cubes too have some interesting patterns:
23 = 8 ∴ 23 – 13
33 = 27 ∴ 33 – 23
43 = 64 ∴ 43 – 33
.
.
.
93 = 729 ∴ 93 – 83
Also,
=7 =1+1×6=1+2× 1× 3
= 19 = 1 + 1 × 6 + 2 × 6 = 1 + 3 × 2 × 3
= 37 = 1 + 1 × 6 + 2 × 6 + 3 × 6 = 1 + 4 × 3 × 3
= 217 = 1 + 1 × 6 + 2 × 6 + . . . + 8 × 6
=1+9× 8× 3
1 = 13
1 + 7 = 23
1 + 7 + 19 = 33
1 + 7 + 19 + 37 = 43
.
.
.
1 + 7 + 19 + . . . + 217 = 93
Note that 23 is the sum of first 2 numbers of 1, 7, 19, 37, . . . . Similarly, 33, 43, . . .,
9 are the sums of first 3, 4, . . . , 9 numbers respectively out of 1, 7, 19, . . ., 217. These
numbers may be obtained by putting n = 1, 2, 3, ... in 1 + n × (n – 1) × 3.
Thus, to find the cube root of a perfect cube, we go on subtracting
1( = 1 + 1 × 0 × 3), 7 (= 1 + 2 × 1 × 3), 19 (= 1 + 3 × 2 × 3), 37 (= 1 + 4 × 3 × 3), . . .
etc., from the given number till we get a zero. The number of times the subtraction is
carried out, gives the cube root. For example,
3
CUBES AND CUBE ROOTS
35
216 – 1 = 215, 215 – 7 = 208, 208 – 19 = 189, 189 – 37 = 152, 152 – 61 = 91,
91 – 91 = 0.
Since we have subtracted six times to get 0, therefore 3 216 = 6.
This method may be used to find the cube roots of small numbers. This method may
also be used to find the smallest number to be added or subtracted to make a non-perfect
cube a perfect cube.
Example 6 : Examine if 400 is a perfect cube. If not, then find the smallest number that
must be subtracted from 400 to obtain a perfect cube.
Solution : 400 – 1 = 399, 399 – 7 = 392, 392 – 19 = 373, 373 – 37 = 336,
336 – 61 = 275, 275 – 91 = 184, 184 – 127 = 57
The next number to be subtracted is 169 which is greater than 57. Therefore, the
process of successive subtraction does not give zero. Hence, 400 is not a perfect cube. If
we subtract 57 from 400, the above process will give zero after 7 successive subtractions.
Therefore, 400 – 57 = 73.
Thus, 57 is the required number (and 343 is the resulting perfect cube). Similarly, if
112 is added to 400, the sum 512 is a perfect cube. (Why ?)
2.6 Cube Root Using Units Digit
We shall now describe a method that can be used to find cube roots of perfect cubes
having at the most six digits. By looking at Table 2.2, we observe that the cube of a
number ending in 0, 1, 4, 5, 6 and 9 ends in 0, 1, 4, 5, 6 and 9 respectively. However, the
cube of a number ending in 2 ends in 8 and vice versa. Similarly, the cube of a number
ending in 3 or 7 ends in 7 or 3 respectively. Thus, by looking at the units digit of a perfect
cube number, we can determine the units digit of its cube root.
Now consider a number which is a perfect cube and has at the most six digits. The
cube root of such a number has at the most two digits, because the least seven digit
number is 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 at the units place of the perfect cube and determine the digit at
the units place in the cube root as discussed above.
Step 2 : Strike out from the right, last three (i.e., units, tens and hundreds) digits of the
number. If nothing is left, we stop. The digit in Step 1 is the cube root.
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MATHEMATICS
Step 3 : Consider the number left from Step 2. Find the largest single digit number whose
cube is less than or equal to this left over number. This is the tens digit of the cube root.
Example 7 : Find the cube roots of the following numbers :
(i) 512
(ii) 2197
(iii) 117649
(iv) 636056
Solution : (i) 512 : The units digit of 512 is 2. Therefore, the digit at the units place in the
cube root is 8. Since no number is left after striking out the units, tens and hundreds digits
of the number, the required cube root is 8.
(ii) 2197 : Here, units digit is 7. Therefore, units digit of the cube root is 3. After striking out
the last three digits from the right, we are left with the number 2. Now 1 is the largest
number whose cube is less than 2. Therefore, the tens digit is 1.
Thus, the required cube root is 13.
(iii) 117649 : Here, units digit is 9. Therefore, the units digit of the cube root is 9. Striking
out the last three digits from the right, the number left is 117. Now 43 = 64 < 117 and
53 = 125 > 117.
Hence, the tens digit of the cube root is 4.
3
117649 = 49
∴
(iv) 636056 : Here, units digit of the cube root is 6. (Why ?)
Also, 83 < 636 and 93 > 636.
Hence, tens digit of the cube root is 8
3
∴
636056 = 86
2.7 Cube Root by Prime Factorisation
We have already observed that in the prime factorisation of a perfect cube, primes occur
in triples. We, therefore, can find 3 n using the following algorithm :
1.
Find the prime factorisation of n .
2.
Group the factors in triples such that all three factors in each triple are the same.
3.
If some prime factors are left ungrouped, the number n is not a perfect cube and the
process stops.
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.
CUBES AND CUBE ROOTS
Example 8 : Find the cube root of :
(i) 91125
(ii) 531441
(iii) 551368
Solution : (i) 91125 = 5 × 5 × 5 × 3 × 3 × 3 × 3 × 3 × 3
∴
3
91125 = 5 ´ 3 ´ 3
= 45
(ii) 531441 = 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3
∴
3
531441 = 3 ´ 3 ´ 3 ´ 3
= 81
(iii) 551368 = 2 × 2 × 2 × 41 × 41 × 41
∴
3
551368 = 2 ´ 41
= 82
37
2 551368
2 275684
2 137842
41 68921
41
1681
41
5 91125
5 18225
5 3645
3
729
3
243
3
81
3
27
3
9
3
3 531441
3 177147
3 59049
3 19683
3
6561
3
2187
3
729
3
243
3
81
3
27
3
9
3
2.8 Cube Roots of Negative Numbers
Consider the perfect cubes 27 ( = 33) and 343 (= 73). We know that 27 × 343 = 9261 is
also a perfect cube, because a3 × b3 = (a × b)3 holds for integers a and b. Now we find the
cube root of 9261 and observe that
i.e.,
3
27 ´ 343 =
3
27 ´
3
343
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MATHEMATICS
There is nothing special about 27 or 343. In fact, for any two perfect cubes x and y, we
have
x´ y =
3
3
x ´
3
y
i.e., cube root of a product of two perfect cubes is the product of their cube roots.
Making use of this result, we find the cube roots of negative integers. For a positive
integer m
–m=–1× m
∴
3
However,
∴
-m =
3
3
3
-1 ´
3
m
3
-1 = - 1, because (–1) = – 1
-m = - 3 m
Example 9 : Find the cube roots of (i) – 125, (ii) – 343 and (iii) – 2197.
Solution : (i) 3 -125 = - 3 125 = – 5
( Q 53 = 125)
(ii) 3 -343 = - 3 343 = – 7
( Q 73 = 343)
(iii) 3 -2197 = - 3 2197 = – 13
( Q 133 = 2197)
2.9 Cube Roots of Rational Numbers
Like the cube root of the product of two perfect cubes, we have the following result for
the cube root of the quotient of two perfect cubes :
x
If x and y (≠ 0) are perfect cubes, then 3 y =
3
3
x
, i.e., the cube root of the quotient
y
of two perfect cubes is the quotient of their cube roots.
Note that 3 x and 3 y are integers and 3 y ¹ 0. Hence,
3
3
x
is a rational number.
y
Thus, the cube root of a rational number whose numerator and denominator are perfect
cubes, is also a rational number. The numerator of the cube root is the cube root of the
numerator of the given number and the denominator of the cube root is the cube root of
the denominator of the given number.
CUBES AND CUBE ROOTS
Example 10 : Find the cube roots of (i)
Solution : (i)
3
343
=
125
3
343
=
3
125
3
3
39
-2197
-27
343 ,
(ii)
and (iii)
.
1331
512
125
7´7´7
7
=
5
5´5´5
(ii)
(iii)
3
- 3 2197
-2197
- 3 13 ´ 13 ´ 13
-13
13
= 3
= 3
=
=1331
11
11
1331
11 ´ 11 ´ 11
EXERCISE 2.2
1.
Find the cube roots of the following numbers by successive subtraction of numbers
1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, . . . :
(i) 64
2.
(ii) 512
(iii) 1728
Using the method of Question 1, examine if the following numbers are perfect cubes :
(i) 130
(ii) 345
(iii) 792
(iv) 1331
3.
Find the smallest number that must be subtracted from those numbers in Question 2
which are not perfect cubes so as to make them perfect cubes. What are the
corresponding cube roots?
4.
Find the units digits of the cube roots of the following numbers :
(i) 226981
(ii) 13824
(iii) 571787
(iv) 175616
5.
Find the tens digit of the cube root of each of the numbers in Question 4.
6.
Find the cube roots of the following numbers by finding their units and tens digits :
(i) 389017
7.
(iii) 110592
(iv) 46656
Find the cube roots of the following numbers using prime factorisation :
(i) 250047
8.
(ii) 91125
(ii) 438976
(iii) 592704
(iv) 614125
Find the cube roots of
(i) – 226981 (ii) – 13824 (iii) – 571787 (iv) – 175616
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MATHEMATICS
9.
Find the cube roots of the numbers 2460375, 20346417, 210644875, 57066625 using
the fact that :
(i) 2460375 = 3375 × 729
(ii) 20346417 = 9261 × 2197
(iii) 210644875 = 42875 × 4913
(iv) 57066625 = 166375 × 343
[Hint : a3 b3 = (ab)3]
10. Find the cube roots of :
343
729
3375
9261
(ii)
(iii)
(iv)
(i)
166375
2197
4913
42875
11. Examine, whether or not each of the following numbers has a cube root. If not, find
the smallest number by which the number must be multiplied so that the product
has a cube root.
(i) 3087
(ii) 33275
(iii) 120393
12. Find the smallest number by which the numbers in Question 11 must be divided so that
the quotient has a cube root.
Things to Remember
1.
A number n is a perfect cube, if there is an integer m such that n = m3.
2.
If n is a perfect cube and n = m3, then m is a cube root of n. A cube root of n is
written as
3
n.
3.
The units digit of the cube root of a perfect cube can be determined with the help
of the units digit of the perfect cube.
4.
The cube root of a perfect cube can be obtained by prime factorisation of the
number.
5.
The cube root of a product of two perfect cubes is the product of the cube roots
of the perfect cubes, i.e.,
3
6.
3
a . 3 b , where a and b are perfect cubes.
The cube root of a quotient of two perfect cubes is the quotient of their cube
roots, i.e.,
3
7.
ab =
a
=
b
3
3
a
, b ≠ 0, where a and b are perfect cubes.
b
The cube root of a negative perfect cube is negative.