Exercise 4.4 - Cubes & Cube Roots RD Sharma Solutions | Mathematics (Maths) Class 8 PDF Download

 Page 1


Question:38
Find the cube roots of each of the following integers:
i -125
ii -5832
iii -2744000
iv -753571
v -32768
Solution:
i
We have:
3
v -125 = -
3
v 125 = -
3
v 5 ×5 ×5 = -5
ii
We have:
3
v -5832 = -
3
v 5832
To find the cube root of 5832, we use the method of unit digits.
Let us consider the number 5832.
The unit digit is 2; therefore the unit digit in the cube root of 5832 will be 8.
After striking out the units, tens and hundreds digits of the given number, we are left with 5.
Now, 1 is the largest number whose cube is less than or equal to 5.
Therefore, the tens digit of the cube root of 5832 is 1.
?
3
v 5832 = 18
?
3
v -5832 = -
3
v 5832 = -18
iii
We have:
3
v -2744000 = -
3
v 2744000
To find the cube root of 2744000, we use the method of factorisation.
On factorising 2744000 into prime factors, we get:
2744000 = 2 ×2 ×2 ×2 ×2 ×2 ×5 ×5 ×5 ×7 ×7 ×7
On grouping the factors in triples of equal factors, we get:
2744000 = {2 ×2 ×2}×{2 ×2 ×2}×{5 ×5 ×5}×{7 ×7 ×7}
It is evident that the prime factors of 2744000 can be grouped into triples of equal factors and no factor is left over.
Now, collect one factor from each triplet and multiply; we get: 
2 ×2 ×5 ×7 = 140 
This implies that 2744000 is a cube of 140.
Hence, 
3
v -2744000 = -
3
v 2744000 = -140
iv
We have:
3
v -753571 = -
3
v 753571
To find the cube root of 753571, we use the method of unit digits.
Let us consider the number 753571.
The unit digit is 1; therefore the unit digit in the cube root of 753571 will be 1.
After striking out the units, tens and hundreds digits of the given number, we are left with 753.
Now, 9 is the largest number whose cube is less than or equal to 753 (9
3
< 753 < 10
3
).
Therefore, the tens digit of the cube root 753571 is 9.
? 
3
v 753571 = 91
? 
3
v -753571 = -
3
v 753571 = -91
v
We have:
3
v -32768 = -
3
v 32768
To find the cube root of 32768, we use the method of unit digits.
Let us consider the number 32768.
The unit digit is 8; therefore, the unit digit in the cube root of 32768 will be 2.
After striking out the units, tens and hundreds digits of the given number, we are left with 32.
Now, 3 is the largest number whose cube is less than or equal to 32 (3
3
< 32 < 4
3
).
Therefore, the tens digit of the cube root 32768 is 3.
? 
3
v 32768 = 32 
 
? 
3
v -32768 = -
3
v 32768 = -32
Question:39
Show that:
i 
3
v 27 ×
3
v 64 =
3
v 27 ×64
ii 
3
v 64 ×729 =
3
v 64 ×
3
v 729
iv 
3
v -125 ×216 =
3
v -125 ×
3
v 216
v 
3
v -125 -1000 =
3
v -125 ×
3
v -1000
Page 2


Question:38
Find the cube roots of each of the following integers:
i -125
ii -5832
iii -2744000
iv -753571
v -32768
Solution:
i
We have:
3
v -125 = -
3
v 125 = -
3
v 5 ×5 ×5 = -5
ii
We have:
3
v -5832 = -
3
v 5832
To find the cube root of 5832, we use the method of unit digits.
Let us consider the number 5832.
The unit digit is 2; therefore the unit digit in the cube root of 5832 will be 8.
After striking out the units, tens and hundreds digits of the given number, we are left with 5.
Now, 1 is the largest number whose cube is less than or equal to 5.
Therefore, the tens digit of the cube root of 5832 is 1.
?
3
v 5832 = 18
?
3
v -5832 = -
3
v 5832 = -18
iii
We have:
3
v -2744000 = -
3
v 2744000
To find the cube root of 2744000, we use the method of factorisation.
On factorising 2744000 into prime factors, we get:
2744000 = 2 ×2 ×2 ×2 ×2 ×2 ×5 ×5 ×5 ×7 ×7 ×7
On grouping the factors in triples of equal factors, we get:
2744000 = {2 ×2 ×2}×{2 ×2 ×2}×{5 ×5 ×5}×{7 ×7 ×7}
It is evident that the prime factors of 2744000 can be grouped into triples of equal factors and no factor is left over.
Now, collect one factor from each triplet and multiply; we get: 
2 ×2 ×5 ×7 = 140 
This implies that 2744000 is a cube of 140.
Hence, 
3
v -2744000 = -
3
v 2744000 = -140
iv
We have:
3
v -753571 = -
3
v 753571
To find the cube root of 753571, we use the method of unit digits.
Let us consider the number 753571.
The unit digit is 1; therefore the unit digit in the cube root of 753571 will be 1.
After striking out the units, tens and hundreds digits of the given number, we are left with 753.
Now, 9 is the largest number whose cube is less than or equal to 753 (9
3
< 753 < 10
3
).
Therefore, the tens digit of the cube root 753571 is 9.
? 
3
v 753571 = 91
? 
3
v -753571 = -
3
v 753571 = -91
v
We have:
3
v -32768 = -
3
v 32768
To find the cube root of 32768, we use the method of unit digits.
Let us consider the number 32768.
The unit digit is 8; therefore, the unit digit in the cube root of 32768 will be 2.
After striking out the units, tens and hundreds digits of the given number, we are left with 32.
Now, 3 is the largest number whose cube is less than or equal to 32 (3
3
< 32 < 4
3
).
Therefore, the tens digit of the cube root 32768 is 3.
? 
3
v 32768 = 32 
 
? 
3
v -32768 = -
3
v 32768 = -32
Question:39
Show that:
i 
3
v 27 ×
3
v 64 =
3
v 27 ×64
ii 
3
v 64 ×729 =
3
v 64 ×
3
v 729
iv 
3
v -125 ×216 =
3
v -125 ×
3
v 216
v 
3
v -125 -1000 =
3
v -125 ×
3
v -1000
Solution:
i
LHS = 
3
v 27 ×
3
v 64 =
3
v 3 ×3 ×3 ×
3
v 4 ×4 ×4 = 3 ×4 = 12
RHS = 
3
v 27 ×64 =
3
v 3 ×3 ×3 ×4 ×4 ×4 =
3
v {3 ×3 ×3}×{4 ×4 ×4} = 3 ×4 = 12
Because LHS is equal to RHS, the equation is true.
ii
LHS = 
3
v 64 ×729 =
3
v 4 ×4 ×4 ×9 ×9 ×9 =
3
v {4 ×4 ×4}×{9 ×9 ×9} = 4 ×9 = 36
RHS = 
3
v 64 ×
3
v 729 =
3
v 4 ×4 ×4 ×
3
v 9 ×9 ×9 = 4 ×9 = 36
Because LHS is equal to RHS, the equation is true.
iii
LHS = 
3
v -125 ×216 =
3
v -5 × -5 × -5 ×{2 ×2 ×2 ×3 ×3 ×3} =
3
v {-5 × -5 × -5}×{2 ×2 ×2}×{3 ×3 ×3} = -5 ×2 ×3 = -30
RHS = 
3
v -125 ×
3
v 216 =
3
v -5 × -5 × -5 ×
3
v {2 ×2 ×2}×{3 ×3 ×3} = -5 ×(2 ×3) = -30
Because LHS is equal to RHS, the equation is true.
iv
LHS = 
3
v -125 × -1000 =
3
v -5 × -5 × -5 × -10 × -10 × -10 =
3
v {-5 × -5 × -5}×{-10 × -10 × -10} = -5 × -10 = 50
RHS = 
3
v -125 ×
3
v -1000 =
3
v -5 × -5 × -5 ×
3
v {-10 × -10 × -10} = -5 × -10 = 50
Because LHS is equal to RHS, the equation is true.
Question:40
Find the cube root of each of the following numbers:
i 8 × 125
ii -1728 × 216
iii -27 × 2744
iv -729 × -15625
Solution:
Property:
For any two integers a and b, 
3
v ab =
3
v a ×
3
v b
i
From the above property, we have:
3
v 8 ×125 =
3
v 8 ×
3
v 125 =
3
v 2 ×2 ×2 ×
3
v 5 ×5 ×5 = 2 ×5 = 10
ii
From the above property, we have: ? 3
v -1728 ×216 =
3
v -1728 ×
3
v 216
= -
3
v 1728 ×
3
v 216    (For any positive integer x, 
3
v -x = -
3
v x )
Cube root using units digit:
Let us consider the number 1728.
The unit digit is 8; therefore, the unit digit in the cube root of 1728 will be 2.
After striking out the units, tens and hundreds digits of the given number, we are left with 1.
Now, 1 is the largest number whose cube is less than or equal to 1.
Therefore, the tens digit of the cube root of 1728 is 1.
? 
3
v 1728 = 12
On factorising 216 into prime factors, we get:
216 = 2 ×2 ×2 ×3 ×3 ×3
On grouping the factors in triples of equal factors, we get:
216 = {2 ×2 ×2}×{3 ×3 ×3}
Now, taking one factor from each triple, we get:
3
v 216 = 2 ×3 = 6
Thus
 
3
v -1728 ×216 = -
3
v 1728 ×
3
v 216 = -12 ×6 = -72
iii
From the above property, we have: ? 3
v -27 ×2744 =
3
v -27 ×
3
v 2744
= -
3
v 27 ×
3
v 2744    (For any positive integer x, 
3
v -x = -
3
v x )
Cube root using units digit:
Let us consider the number 2744.
The unit digit is 4; therefore, the unit digit in the cube root of 2744 will be 4.
After striking out the units, tens, and hundreds digits of the given number, we are left with 2.
Now, 1 is the largest number whose cube is less than or equal to 2.
Therefore, the tens digit of the cube root of 2744 is 1.
? 
3
v 2744 = 14
Thus
 
Page 3


Question:38
Find the cube roots of each of the following integers:
i -125
ii -5832
iii -2744000
iv -753571
v -32768
Solution:
i
We have:
3
v -125 = -
3
v 125 = -
3
v 5 ×5 ×5 = -5
ii
We have:
3
v -5832 = -
3
v 5832
To find the cube root of 5832, we use the method of unit digits.
Let us consider the number 5832.
The unit digit is 2; therefore the unit digit in the cube root of 5832 will be 8.
After striking out the units, tens and hundreds digits of the given number, we are left with 5.
Now, 1 is the largest number whose cube is less than or equal to 5.
Therefore, the tens digit of the cube root of 5832 is 1.
?
3
v 5832 = 18
?
3
v -5832 = -
3
v 5832 = -18
iii
We have:
3
v -2744000 = -
3
v 2744000
To find the cube root of 2744000, we use the method of factorisation.
On factorising 2744000 into prime factors, we get:
2744000 = 2 ×2 ×2 ×2 ×2 ×2 ×5 ×5 ×5 ×7 ×7 ×7
On grouping the factors in triples of equal factors, we get:
2744000 = {2 ×2 ×2}×{2 ×2 ×2}×{5 ×5 ×5}×{7 ×7 ×7}
It is evident that the prime factors of 2744000 can be grouped into triples of equal factors and no factor is left over.
Now, collect one factor from each triplet and multiply; we get: 
2 ×2 ×5 ×7 = 140 
This implies that 2744000 is a cube of 140.
Hence, 
3
v -2744000 = -
3
v 2744000 = -140
iv
We have:
3
v -753571 = -
3
v 753571
To find the cube root of 753571, we use the method of unit digits.
Let us consider the number 753571.
The unit digit is 1; therefore the unit digit in the cube root of 753571 will be 1.
After striking out the units, tens and hundreds digits of the given number, we are left with 753.
Now, 9 is the largest number whose cube is less than or equal to 753 (9
3
< 753 < 10
3
).
Therefore, the tens digit of the cube root 753571 is 9.
? 
3
v 753571 = 91
? 
3
v -753571 = -
3
v 753571 = -91
v
We have:
3
v -32768 = -
3
v 32768
To find the cube root of 32768, we use the method of unit digits.
Let us consider the number 32768.
The unit digit is 8; therefore, the unit digit in the cube root of 32768 will be 2.
After striking out the units, tens and hundreds digits of the given number, we are left with 32.
Now, 3 is the largest number whose cube is less than or equal to 32 (3
3
< 32 < 4
3
).
Therefore, the tens digit of the cube root 32768 is 3.
? 
3
v 32768 = 32 
 
? 
3
v -32768 = -
3
v 32768 = -32
Question:39
Show that:
i 
3
v 27 ×
3
v 64 =
3
v 27 ×64
ii 
3
v 64 ×729 =
3
v 64 ×
3
v 729
iv 
3
v -125 ×216 =
3
v -125 ×
3
v 216
v 
3
v -125 -1000 =
3
v -125 ×
3
v -1000
Solution:
i
LHS = 
3
v 27 ×
3
v 64 =
3
v 3 ×3 ×3 ×
3
v 4 ×4 ×4 = 3 ×4 = 12
RHS = 
3
v 27 ×64 =
3
v 3 ×3 ×3 ×4 ×4 ×4 =
3
v {3 ×3 ×3}×{4 ×4 ×4} = 3 ×4 = 12
Because LHS is equal to RHS, the equation is true.
ii
LHS = 
3
v 64 ×729 =
3
v 4 ×4 ×4 ×9 ×9 ×9 =
3
v {4 ×4 ×4}×{9 ×9 ×9} = 4 ×9 = 36
RHS = 
3
v 64 ×
3
v 729 =
3
v 4 ×4 ×4 ×
3
v 9 ×9 ×9 = 4 ×9 = 36
Because LHS is equal to RHS, the equation is true.
iii
LHS = 
3
v -125 ×216 =
3
v -5 × -5 × -5 ×{2 ×2 ×2 ×3 ×3 ×3} =
3
v {-5 × -5 × -5}×{2 ×2 ×2}×{3 ×3 ×3} = -5 ×2 ×3 = -30
RHS = 
3
v -125 ×
3
v 216 =
3
v -5 × -5 × -5 ×
3
v {2 ×2 ×2}×{3 ×3 ×3} = -5 ×(2 ×3) = -30
Because LHS is equal to RHS, the equation is true.
iv
LHS = 
3
v -125 × -1000 =
3
v -5 × -5 × -5 × -10 × -10 × -10 =
3
v {-5 × -5 × -5}×{-10 × -10 × -10} = -5 × -10 = 50
RHS = 
3
v -125 ×
3
v -1000 =
3
v -5 × -5 × -5 ×
3
v {-10 × -10 × -10} = -5 × -10 = 50
Because LHS is equal to RHS, the equation is true.
Question:40
Find the cube root of each of the following numbers:
i 8 × 125
ii -1728 × 216
iii -27 × 2744
iv -729 × -15625
Solution:
Property:
For any two integers a and b, 
3
v ab =
3
v a ×
3
v b
i
From the above property, we have:
3
v 8 ×125 =
3
v 8 ×
3
v 125 =
3
v 2 ×2 ×2 ×
3
v 5 ×5 ×5 = 2 ×5 = 10
ii
From the above property, we have: ? 3
v -1728 ×216 =
3
v -1728 ×
3
v 216
= -
3
v 1728 ×
3
v 216    (For any positive integer x, 
3
v -x = -
3
v x )
Cube root using units digit:
Let us consider the number 1728.
The unit digit is 8; therefore, the unit digit in the cube root of 1728 will be 2.
After striking out the units, tens and hundreds digits of the given number, we are left with 1.
Now, 1 is the largest number whose cube is less than or equal to 1.
Therefore, the tens digit of the cube root of 1728 is 1.
? 
3
v 1728 = 12
On factorising 216 into prime factors, we get:
216 = 2 ×2 ×2 ×3 ×3 ×3
On grouping the factors in triples of equal factors, we get:
216 = {2 ×2 ×2}×{3 ×3 ×3}
Now, taking one factor from each triple, we get:
3
v 216 = 2 ×3 = 6
Thus
 
3
v -1728 ×216 = -
3
v 1728 ×
3
v 216 = -12 ×6 = -72
iii
From the above property, we have: ? 3
v -27 ×2744 =
3
v -27 ×
3
v 2744
= -
3
v 27 ×
3
v 2744    (For any positive integer x, 
3
v -x = -
3
v x )
Cube root using units digit:
Let us consider the number 2744.
The unit digit is 4; therefore, the unit digit in the cube root of 2744 will be 4.
After striking out the units, tens, and hundreds digits of the given number, we are left with 2.
Now, 1 is the largest number whose cube is less than or equal to 2.
Therefore, the tens digit of the cube root of 2744 is 1.
? 
3
v 2744 = 14
Thus
 
3
v -27 ×2744 = -
3
v 27 ×
3
v 2744 = -3 ×14 = -42
iv
From the above property, we have: ? 3
v -729 × -15625 =
3
v -729 ×
3
v -15625
= -
3
v 729 × -
3
v 15625    (For any positive integer x, 
3
v -x = -
3
v x )
Cube root using units digit:
Let us consider the number 15625.
The unit digit is 5; therefore, the unit digit in the cube root of 15625 will be 5.
After striking out the units, tens and hundreds digits of the given number, we are left with 15.
Now, 2 is the largest number whose cube is less than or equal to 15 ( 2
3
< 15 < 3
3
.
Therefore, the tens digit of the cube root of 15625 is 2.
? 
3
v 15625 = 25 
Also
3
v 729 = 9, because 9 ×9 ×9 = 729
Thus
3
v -729 × -15625 = -
3
v 729 × -
3
v 15625 = -9 × -25 = 225
Question:41
Evaluate:
i 
3
v
4
3
×6
3
ii 
3
v 8 ×17 ×17 ×17
iii 
3
v 700 ×2 ×49 ×5
iv 125
3
v
a
6
-
3
v
125 a
6
Solution:
Property:
For any two integers a and b, 
3
v ab =
3
v a ×
3
v b
i From the above property, we have:
3
v
4
3
×6
3
=
3
v
4
3
×
3
v
6
3
= 4 ×6 = 24
ii Use above property and proceed as follows:
3
v 8 ×17 ×17 ×17 =
3
v
2
3
×17
3
=
3
v
2
3
×
3
v
17
3
= 2 ×17 = 34
iii From the above property, we have: ? 3
v 700 ×2 ×49 ×5
=
3
v 2 ×2 ×5 ×5 ×7 ×2 ×7 ×7 ×5   ( ? 700 = 2 ×2 ×5 ×5 ×7 and  49 = 7 ×7)
=
3
v
2
3
×5
3
×7
3
=
3
v
2
3
×
3
v
5
3
×
3
v
7
3
= 2 ×5 ×7 = 70
iv
From the above property, we have: ? 125
3
v
a
6
-
3
v
125a
6
= 125
3
v
a
6
-
3
v 125 ×
6
v
a
6
= 125 ×a
2
- 5 ×a
2
                  ( ? 
3
v
a
6
=
3
v {a ×a ×a}×{a ×a ×a} = a ×a = a
2
 and 
3
v 125 =
3
v 5 ×5 ×5 = 5)
= 125a
2
-5a
2
= 120a
2
Question:42
Find the cube root of each of the following rational numbers:
i 
-125
729
ii 
10648
12167
iii 
-19683
24389
iv 
686
-3456
v 
-39304
-42875
Solution:
i
Let us consider the following rational number:
-125
729
Now
3 -125
729
=
3
v
-125
3
v
729
          ( ? 
3 a
b
=
3
v
a
3
v
b
)
( )
( )
( )
v
v
Page 4


Question:38
Find the cube roots of each of the following integers:
i -125
ii -5832
iii -2744000
iv -753571
v -32768
Solution:
i
We have:
3
v -125 = -
3
v 125 = -
3
v 5 ×5 ×5 = -5
ii
We have:
3
v -5832 = -
3
v 5832
To find the cube root of 5832, we use the method of unit digits.
Let us consider the number 5832.
The unit digit is 2; therefore the unit digit in the cube root of 5832 will be 8.
After striking out the units, tens and hundreds digits of the given number, we are left with 5.
Now, 1 is the largest number whose cube is less than or equal to 5.
Therefore, the tens digit of the cube root of 5832 is 1.
?
3
v 5832 = 18
?
3
v -5832 = -
3
v 5832 = -18
iii
We have:
3
v -2744000 = -
3
v 2744000
To find the cube root of 2744000, we use the method of factorisation.
On factorising 2744000 into prime factors, we get:
2744000 = 2 ×2 ×2 ×2 ×2 ×2 ×5 ×5 ×5 ×7 ×7 ×7
On grouping the factors in triples of equal factors, we get:
2744000 = {2 ×2 ×2}×{2 ×2 ×2}×{5 ×5 ×5}×{7 ×7 ×7}
It is evident that the prime factors of 2744000 can be grouped into triples of equal factors and no factor is left over.
Now, collect one factor from each triplet and multiply; we get: 
2 ×2 ×5 ×7 = 140 
This implies that 2744000 is a cube of 140.
Hence, 
3
v -2744000 = -
3
v 2744000 = -140
iv
We have:
3
v -753571 = -
3
v 753571
To find the cube root of 753571, we use the method of unit digits.
Let us consider the number 753571.
The unit digit is 1; therefore the unit digit in the cube root of 753571 will be 1.
After striking out the units, tens and hundreds digits of the given number, we are left with 753.
Now, 9 is the largest number whose cube is less than or equal to 753 (9
3
< 753 < 10
3
).
Therefore, the tens digit of the cube root 753571 is 9.
? 
3
v 753571 = 91
? 
3
v -753571 = -
3
v 753571 = -91
v
We have:
3
v -32768 = -
3
v 32768
To find the cube root of 32768, we use the method of unit digits.
Let us consider the number 32768.
The unit digit is 8; therefore, the unit digit in the cube root of 32768 will be 2.
After striking out the units, tens and hundreds digits of the given number, we are left with 32.
Now, 3 is the largest number whose cube is less than or equal to 32 (3
3
< 32 < 4
3
).
Therefore, the tens digit of the cube root 32768 is 3.
? 
3
v 32768 = 32 
 
? 
3
v -32768 = -
3
v 32768 = -32
Question:39
Show that:
i 
3
v 27 ×
3
v 64 =
3
v 27 ×64
ii 
3
v 64 ×729 =
3
v 64 ×
3
v 729
iv 
3
v -125 ×216 =
3
v -125 ×
3
v 216
v 
3
v -125 -1000 =
3
v -125 ×
3
v -1000
Solution:
i
LHS = 
3
v 27 ×
3
v 64 =
3
v 3 ×3 ×3 ×
3
v 4 ×4 ×4 = 3 ×4 = 12
RHS = 
3
v 27 ×64 =
3
v 3 ×3 ×3 ×4 ×4 ×4 =
3
v {3 ×3 ×3}×{4 ×4 ×4} = 3 ×4 = 12
Because LHS is equal to RHS, the equation is true.
ii
LHS = 
3
v 64 ×729 =
3
v 4 ×4 ×4 ×9 ×9 ×9 =
3
v {4 ×4 ×4}×{9 ×9 ×9} = 4 ×9 = 36
RHS = 
3
v 64 ×
3
v 729 =
3
v 4 ×4 ×4 ×
3
v 9 ×9 ×9 = 4 ×9 = 36
Because LHS is equal to RHS, the equation is true.
iii
LHS = 
3
v -125 ×216 =
3
v -5 × -5 × -5 ×{2 ×2 ×2 ×3 ×3 ×3} =
3
v {-5 × -5 × -5}×{2 ×2 ×2}×{3 ×3 ×3} = -5 ×2 ×3 = -30
RHS = 
3
v -125 ×
3
v 216 =
3
v -5 × -5 × -5 ×
3
v {2 ×2 ×2}×{3 ×3 ×3} = -5 ×(2 ×3) = -30
Because LHS is equal to RHS, the equation is true.
iv
LHS = 
3
v -125 × -1000 =
3
v -5 × -5 × -5 × -10 × -10 × -10 =
3
v {-5 × -5 × -5}×{-10 × -10 × -10} = -5 × -10 = 50
RHS = 
3
v -125 ×
3
v -1000 =
3
v -5 × -5 × -5 ×
3
v {-10 × -10 × -10} = -5 × -10 = 50
Because LHS is equal to RHS, the equation is true.
Question:40
Find the cube root of each of the following numbers:
i 8 × 125
ii -1728 × 216
iii -27 × 2744
iv -729 × -15625
Solution:
Property:
For any two integers a and b, 
3
v ab =
3
v a ×
3
v b
i
From the above property, we have:
3
v 8 ×125 =
3
v 8 ×
3
v 125 =
3
v 2 ×2 ×2 ×
3
v 5 ×5 ×5 = 2 ×5 = 10
ii
From the above property, we have: ? 3
v -1728 ×216 =
3
v -1728 ×
3
v 216
= -
3
v 1728 ×
3
v 216    (For any positive integer x, 
3
v -x = -
3
v x )
Cube root using units digit:
Let us consider the number 1728.
The unit digit is 8; therefore, the unit digit in the cube root of 1728 will be 2.
After striking out the units, tens and hundreds digits of the given number, we are left with 1.
Now, 1 is the largest number whose cube is less than or equal to 1.
Therefore, the tens digit of the cube root of 1728 is 1.
? 
3
v 1728 = 12
On factorising 216 into prime factors, we get:
216 = 2 ×2 ×2 ×3 ×3 ×3
On grouping the factors in triples of equal factors, we get:
216 = {2 ×2 ×2}×{3 ×3 ×3}
Now, taking one factor from each triple, we get:
3
v 216 = 2 ×3 = 6
Thus
 
3
v -1728 ×216 = -
3
v 1728 ×
3
v 216 = -12 ×6 = -72
iii
From the above property, we have: ? 3
v -27 ×2744 =
3
v -27 ×
3
v 2744
= -
3
v 27 ×
3
v 2744    (For any positive integer x, 
3
v -x = -
3
v x )
Cube root using units digit:
Let us consider the number 2744.
The unit digit is 4; therefore, the unit digit in the cube root of 2744 will be 4.
After striking out the units, tens, and hundreds digits of the given number, we are left with 2.
Now, 1 is the largest number whose cube is less than or equal to 2.
Therefore, the tens digit of the cube root of 2744 is 1.
? 
3
v 2744 = 14
Thus
 
3
v -27 ×2744 = -
3
v 27 ×
3
v 2744 = -3 ×14 = -42
iv
From the above property, we have: ? 3
v -729 × -15625 =
3
v -729 ×
3
v -15625
= -
3
v 729 × -
3
v 15625    (For any positive integer x, 
3
v -x = -
3
v x )
Cube root using units digit:
Let us consider the number 15625.
The unit digit is 5; therefore, the unit digit in the cube root of 15625 will be 5.
After striking out the units, tens and hundreds digits of the given number, we are left with 15.
Now, 2 is the largest number whose cube is less than or equal to 15 ( 2
3
< 15 < 3
3
.
Therefore, the tens digit of the cube root of 15625 is 2.
? 
3
v 15625 = 25 
Also
3
v 729 = 9, because 9 ×9 ×9 = 729
Thus
3
v -729 × -15625 = -
3
v 729 × -
3
v 15625 = -9 × -25 = 225
Question:41
Evaluate:
i 
3
v
4
3
×6
3
ii 
3
v 8 ×17 ×17 ×17
iii 
3
v 700 ×2 ×49 ×5
iv 125
3
v
a
6
-
3
v
125 a
6
Solution:
Property:
For any two integers a and b, 
3
v ab =
3
v a ×
3
v b
i From the above property, we have:
3
v
4
3
×6
3
=
3
v
4
3
×
3
v
6
3
= 4 ×6 = 24
ii Use above property and proceed as follows:
3
v 8 ×17 ×17 ×17 =
3
v
2
3
×17
3
=
3
v
2
3
×
3
v
17
3
= 2 ×17 = 34
iii From the above property, we have: ? 3
v 700 ×2 ×49 ×5
=
3
v 2 ×2 ×5 ×5 ×7 ×2 ×7 ×7 ×5   ( ? 700 = 2 ×2 ×5 ×5 ×7 and  49 = 7 ×7)
=
3
v
2
3
×5
3
×7
3
=
3
v
2
3
×
3
v
5
3
×
3
v
7
3
= 2 ×5 ×7 = 70
iv
From the above property, we have: ? 125
3
v
a
6
-
3
v
125a
6
= 125
3
v
a
6
-
3
v 125 ×
6
v
a
6
= 125 ×a
2
- 5 ×a
2
                  ( ? 
3
v
a
6
=
3
v {a ×a ×a}×{a ×a ×a} = a ×a = a
2
 and 
3
v 125 =
3
v 5 ×5 ×5 = 5)
= 125a
2
-5a
2
= 120a
2
Question:42
Find the cube root of each of the following rational numbers:
i 
-125
729
ii 
10648
12167
iii 
-19683
24389
iv 
686
-3456
v 
-39304
-42875
Solution:
i
Let us consider the following rational number:
-125
729
Now
3 -125
729
=
3
v
-125
3
v
729
          ( ? 
3 a
b
=
3
v
a
3
v
b
)
( )
( )
( )
v
v
=
-
3
v
125
3
v
729
          ( ? 
3
v -a = -
3
v a )
= -
5
9
                    ( ? 729 = 9 ×9 ×9 and 125 = 5 ×5 ×5)
ii
Let us consider the following rational number:
10648
12167
Now
3 10648
12167
=
3
v
10648
3
v
12167
          ( ? 
3 a
b
=
3
v
a
3
v
b
)
Cube root by factors:
On factorising 10648 into prime factors, we get:
10648 = 2 ×2 ×2 ×11 ×11 ×11
On grouping the factors in triples of equal factors, we get:
10648 = {2 ×2 ×2}×{11 ×11 ×11}
Now, taking one factor from each triple, we get:
3
v 10648 = 2 ×11 = 22
Also
On factorising 12167 into prime factors, we get:
12167 = 23 ×23 ×23
On grouping the factors in triples of equal factors, we get:
12167 = {23 ×23 ×23}
Now, taking one factor from the triple, we get:
3
v 12167 = 23
Now
3 10648
12167
=
3
v
10648
3
v
12167
=
22
23
iii
Let us consider the following rational number:
-19683
24389
Now,
3 -19683
24389
=
3
v
-19683
3
v
24389
          ( ? 
3 a
b
=
3
v
a
3
v
b
)
=
-
3
v
19683
3
v
24389
          ( ? 
3
v -a = -
3
v a )
Cube root by factors:
On factorising 19683 into prime factors, we get:
19683 = 3 ×3 ×3 ×3 ×3 ×3 ×3 ×3 ×3
On grouping the factors in triples of equal factors, we get:
19683 = {3 ×3 ×3}×{3 ×3 ×3}×{3 ×3 ×3}
Now, taking one factor from each triple, we get:
3
v 19683 = 3 ×3 ×3 = 27
Also
On factorising 24389 into prime factors, we get:
24389 = 29 ×29 ×29
On grouping the factors in triples of equal factors, we get:
24389 = {29 ×29 ×29}
Now, taking one factor from each triple, we get:
3
v 24389 = 29
Now
3 -19683
24389
v
v
v
v
v
v
Page 5


Question:38
Find the cube roots of each of the following integers:
i -125
ii -5832
iii -2744000
iv -753571
v -32768
Solution:
i
We have:
3
v -125 = -
3
v 125 = -
3
v 5 ×5 ×5 = -5
ii
We have:
3
v -5832 = -
3
v 5832
To find the cube root of 5832, we use the method of unit digits.
Let us consider the number 5832.
The unit digit is 2; therefore the unit digit in the cube root of 5832 will be 8.
After striking out the units, tens and hundreds digits of the given number, we are left with 5.
Now, 1 is the largest number whose cube is less than or equal to 5.
Therefore, the tens digit of the cube root of 5832 is 1.
?
3
v 5832 = 18
?
3
v -5832 = -
3
v 5832 = -18
iii
We have:
3
v -2744000 = -
3
v 2744000
To find the cube root of 2744000, we use the method of factorisation.
On factorising 2744000 into prime factors, we get:
2744000 = 2 ×2 ×2 ×2 ×2 ×2 ×5 ×5 ×5 ×7 ×7 ×7
On grouping the factors in triples of equal factors, we get:
2744000 = {2 ×2 ×2}×{2 ×2 ×2}×{5 ×5 ×5}×{7 ×7 ×7}
It is evident that the prime factors of 2744000 can be grouped into triples of equal factors and no factor is left over.
Now, collect one factor from each triplet and multiply; we get: 
2 ×2 ×5 ×7 = 140 
This implies that 2744000 is a cube of 140.
Hence, 
3
v -2744000 = -
3
v 2744000 = -140
iv
We have:
3
v -753571 = -
3
v 753571
To find the cube root of 753571, we use the method of unit digits.
Let us consider the number 753571.
The unit digit is 1; therefore the unit digit in the cube root of 753571 will be 1.
After striking out the units, tens and hundreds digits of the given number, we are left with 753.
Now, 9 is the largest number whose cube is less than or equal to 753 (9
3
< 753 < 10
3
).
Therefore, the tens digit of the cube root 753571 is 9.
? 
3
v 753571 = 91
? 
3
v -753571 = -
3
v 753571 = -91
v
We have:
3
v -32768 = -
3
v 32768
To find the cube root of 32768, we use the method of unit digits.
Let us consider the number 32768.
The unit digit is 8; therefore, the unit digit in the cube root of 32768 will be 2.
After striking out the units, tens and hundreds digits of the given number, we are left with 32.
Now, 3 is the largest number whose cube is less than or equal to 32 (3
3
< 32 < 4
3
).
Therefore, the tens digit of the cube root 32768 is 3.
? 
3
v 32768 = 32 
 
? 
3
v -32768 = -
3
v 32768 = -32
Question:39
Show that:
i 
3
v 27 ×
3
v 64 =
3
v 27 ×64
ii 
3
v 64 ×729 =
3
v 64 ×
3
v 729
iv 
3
v -125 ×216 =
3
v -125 ×
3
v 216
v 
3
v -125 -1000 =
3
v -125 ×
3
v -1000
Solution:
i
LHS = 
3
v 27 ×
3
v 64 =
3
v 3 ×3 ×3 ×
3
v 4 ×4 ×4 = 3 ×4 = 12
RHS = 
3
v 27 ×64 =
3
v 3 ×3 ×3 ×4 ×4 ×4 =
3
v {3 ×3 ×3}×{4 ×4 ×4} = 3 ×4 = 12
Because LHS is equal to RHS, the equation is true.
ii
LHS = 
3
v 64 ×729 =
3
v 4 ×4 ×4 ×9 ×9 ×9 =
3
v {4 ×4 ×4}×{9 ×9 ×9} = 4 ×9 = 36
RHS = 
3
v 64 ×
3
v 729 =
3
v 4 ×4 ×4 ×
3
v 9 ×9 ×9 = 4 ×9 = 36
Because LHS is equal to RHS, the equation is true.
iii
LHS = 
3
v -125 ×216 =
3
v -5 × -5 × -5 ×{2 ×2 ×2 ×3 ×3 ×3} =
3
v {-5 × -5 × -5}×{2 ×2 ×2}×{3 ×3 ×3} = -5 ×2 ×3 = -30
RHS = 
3
v -125 ×
3
v 216 =
3
v -5 × -5 × -5 ×
3
v {2 ×2 ×2}×{3 ×3 ×3} = -5 ×(2 ×3) = -30
Because LHS is equal to RHS, the equation is true.
iv
LHS = 
3
v -125 × -1000 =
3
v -5 × -5 × -5 × -10 × -10 × -10 =
3
v {-5 × -5 × -5}×{-10 × -10 × -10} = -5 × -10 = 50
RHS = 
3
v -125 ×
3
v -1000 =
3
v -5 × -5 × -5 ×
3
v {-10 × -10 × -10} = -5 × -10 = 50
Because LHS is equal to RHS, the equation is true.
Question:40
Find the cube root of each of the following numbers:
i 8 × 125
ii -1728 × 216
iii -27 × 2744
iv -729 × -15625
Solution:
Property:
For any two integers a and b, 
3
v ab =
3
v a ×
3
v b
i
From the above property, we have:
3
v 8 ×125 =
3
v 8 ×
3
v 125 =
3
v 2 ×2 ×2 ×
3
v 5 ×5 ×5 = 2 ×5 = 10
ii
From the above property, we have: ? 3
v -1728 ×216 =
3
v -1728 ×
3
v 216
= -
3
v 1728 ×
3
v 216    (For any positive integer x, 
3
v -x = -
3
v x )
Cube root using units digit:
Let us consider the number 1728.
The unit digit is 8; therefore, the unit digit in the cube root of 1728 will be 2.
After striking out the units, tens and hundreds digits of the given number, we are left with 1.
Now, 1 is the largest number whose cube is less than or equal to 1.
Therefore, the tens digit of the cube root of 1728 is 1.
? 
3
v 1728 = 12
On factorising 216 into prime factors, we get:
216 = 2 ×2 ×2 ×3 ×3 ×3
On grouping the factors in triples of equal factors, we get:
216 = {2 ×2 ×2}×{3 ×3 ×3}
Now, taking one factor from each triple, we get:
3
v 216 = 2 ×3 = 6
Thus
 
3
v -1728 ×216 = -
3
v 1728 ×
3
v 216 = -12 ×6 = -72
iii
From the above property, we have: ? 3
v -27 ×2744 =
3
v -27 ×
3
v 2744
= -
3
v 27 ×
3
v 2744    (For any positive integer x, 
3
v -x = -
3
v x )
Cube root using units digit:
Let us consider the number 2744.
The unit digit is 4; therefore, the unit digit in the cube root of 2744 will be 4.
After striking out the units, tens, and hundreds digits of the given number, we are left with 2.
Now, 1 is the largest number whose cube is less than or equal to 2.
Therefore, the tens digit of the cube root of 2744 is 1.
? 
3
v 2744 = 14
Thus
 
3
v -27 ×2744 = -
3
v 27 ×
3
v 2744 = -3 ×14 = -42
iv
From the above property, we have: ? 3
v -729 × -15625 =
3
v -729 ×
3
v -15625
= -
3
v 729 × -
3
v 15625    (For any positive integer x, 
3
v -x = -
3
v x )
Cube root using units digit:
Let us consider the number 15625.
The unit digit is 5; therefore, the unit digit in the cube root of 15625 will be 5.
After striking out the units, tens and hundreds digits of the given number, we are left with 15.
Now, 2 is the largest number whose cube is less than or equal to 15 ( 2
3
< 15 < 3
3
.
Therefore, the tens digit of the cube root of 15625 is 2.
? 
3
v 15625 = 25 
Also
3
v 729 = 9, because 9 ×9 ×9 = 729
Thus
3
v -729 × -15625 = -
3
v 729 × -
3
v 15625 = -9 × -25 = 225
Question:41
Evaluate:
i 
3
v
4
3
×6
3
ii 
3
v 8 ×17 ×17 ×17
iii 
3
v 700 ×2 ×49 ×5
iv 125
3
v
a
6
-
3
v
125 a
6
Solution:
Property:
For any two integers a and b, 
3
v ab =
3
v a ×
3
v b
i From the above property, we have:
3
v
4
3
×6
3
=
3
v
4
3
×
3
v
6
3
= 4 ×6 = 24
ii Use above property and proceed as follows:
3
v 8 ×17 ×17 ×17 =
3
v
2
3
×17
3
=
3
v
2
3
×
3
v
17
3
= 2 ×17 = 34
iii From the above property, we have: ? 3
v 700 ×2 ×49 ×5
=
3
v 2 ×2 ×5 ×5 ×7 ×2 ×7 ×7 ×5   ( ? 700 = 2 ×2 ×5 ×5 ×7 and  49 = 7 ×7)
=
3
v
2
3
×5
3
×7
3
=
3
v
2
3
×
3
v
5
3
×
3
v
7
3
= 2 ×5 ×7 = 70
iv
From the above property, we have: ? 125
3
v
a
6
-
3
v
125a
6
= 125
3
v
a
6
-
3
v 125 ×
6
v
a
6
= 125 ×a
2
- 5 ×a
2
                  ( ? 
3
v
a
6
=
3
v {a ×a ×a}×{a ×a ×a} = a ×a = a
2
 and 
3
v 125 =
3
v 5 ×5 ×5 = 5)
= 125a
2
-5a
2
= 120a
2
Question:42
Find the cube root of each of the following rational numbers:
i 
-125
729
ii 
10648
12167
iii 
-19683
24389
iv 
686
-3456
v 
-39304
-42875
Solution:
i
Let us consider the following rational number:
-125
729
Now
3 -125
729
=
3
v
-125
3
v
729
          ( ? 
3 a
b
=
3
v
a
3
v
b
)
( )
( )
( )
v
v
=
-
3
v
125
3
v
729
          ( ? 
3
v -a = -
3
v a )
= -
5
9
                    ( ? 729 = 9 ×9 ×9 and 125 = 5 ×5 ×5)
ii
Let us consider the following rational number:
10648
12167
Now
3 10648
12167
=
3
v
10648
3
v
12167
          ( ? 
3 a
b
=
3
v
a
3
v
b
)
Cube root by factors:
On factorising 10648 into prime factors, we get:
10648 = 2 ×2 ×2 ×11 ×11 ×11
On grouping the factors in triples of equal factors, we get:
10648 = {2 ×2 ×2}×{11 ×11 ×11}
Now, taking one factor from each triple, we get:
3
v 10648 = 2 ×11 = 22
Also
On factorising 12167 into prime factors, we get:
12167 = 23 ×23 ×23
On grouping the factors in triples of equal factors, we get:
12167 = {23 ×23 ×23}
Now, taking one factor from the triple, we get:
3
v 12167 = 23
Now
3 10648
12167
=
3
v
10648
3
v
12167
=
22
23
iii
Let us consider the following rational number:
-19683
24389
Now,
3 -19683
24389
=
3
v
-19683
3
v
24389
          ( ? 
3 a
b
=
3
v
a
3
v
b
)
=
-
3
v
19683
3
v
24389
          ( ? 
3
v -a = -
3
v a )
Cube root by factors:
On factorising 19683 into prime factors, we get:
19683 = 3 ×3 ×3 ×3 ×3 ×3 ×3 ×3 ×3
On grouping the factors in triples of equal factors, we get:
19683 = {3 ×3 ×3}×{3 ×3 ×3}×{3 ×3 ×3}
Now, taking one factor from each triple, we get:
3
v 19683 = 3 ×3 ×3 = 27
Also
On factorising 24389 into prime factors, we get:
24389 = 29 ×29 ×29
On grouping the factors in triples of equal factors, we get:
24389 = {29 ×29 ×29}
Now, taking one factor from each triple, we get:
3
v 24389 = 29
Now
3 -19683
24389
v
v
v
v
v
v
=
3
v
-19683
3
v
24389
=
-
3
v
19683
3
v
24389
=
-27
29
iv
Let us consider the following rational number:
686
-3456
Now
3 686
-3456
 
= -
3 2×7
3
2
7
×3
3
        (686 and 3456 are not perfect cubes; therefore, we simplify it as 
686
3456
 by prime factorisation.)
= -
3 7
3
2
6
×3
3
   
=
-
3
v
7
3
3
v
2
6
×3
3
=
-7
3
v
2
3
×2
3
×3
3
=
-7
2×2×3
=
-7
12
       ( ? 
3 a
b
=
3
v
a
3
v
b
)
v
Let us consider the following rational number:
-39304
-42875
Now
3 -39304
-42875
=
3
v
-39304
3
v
-42875
          ( ? 
3 a
b
=
3
v
a
3
v
b
)
=
-
3
v
39304
-
3
v
42875
          ( ? 
3
v -a = -
3
v a )
Cube root by factors:
On factorising 39304 into prime factors, we get:
39304 = 2 ×2 ×2 ×17 ×17 ×17
On grouping the factors in triples of equal factors, we get:
39304 = {2 ×2 ×2}×{17 ×17 ×17}
Now, taking one factor from each triple, we get:
3
v 39304 = 2 ×17 = 34
Also
On factorising 42875 into prime factors, we get:
42875 = 5 ×5 ×5 ×7 ×7 ×7
On grouping the factors in triples of equal factors, we get:
42875 = {5 ×5 ×5}×{7 ×7 ×7}
Now, taking one factor from each triple, we get:
3
v 42875 = 5 ×7 = 35
Now
3 -39304
-42875
=
3
v
-39304
3
v
-42875
        
=
-
3
v
39304
-
3
v
42875
       
=
-34
-35
=
34
35
Question:43
Find the cube root of each of the following rational numbers:
i 0.001728
ii 0.003375
iii 0.001
iv 1.331
Solution:
i
We have:
0. 001728 =
1728
1000000
? 
3
v 0. 001728 =
3 1728
1000000
=
3
v
1728
3
v
1000000
v
v
v
v
v
v
v
v