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Page 1
Cube and Cube Roots
Points to Remember
? When a given number is multiplied by itself three times, the new number
obtained is called cube of the given number and given number is called cube root
of the new number i.e. a number ‘m’ is the cube root of a number ‘n’ if ?? = ?? ??
or ?? = v ?? ?? . symbol v
?? represents cube root. E.g. ?? ?? = ?? ???? ?? = v ?? ??
? Numbers 1729, 4104, 13832 etc. are called Hardy-Ramanujan numbers. They
can be expressed as the sum of two cubes in two different ways.
???????? = ?? ?? + ????
?? ???? ?? ?? + ????
??
? Cube of a number having 1, 4, 6 or 9 as one’s digit has the same digit at one’s
place.
As ?? ?? = ?? (?? ?? has 1 at its unit place)
?? ?? = ???? (?? ?? has 4 at its unit place)
?? ?? = ?????? (?? ?? has 9 at its unit place)
? If a number has 3 at its one’s place, then its cube has 7 at its one’s place, also if a
number has 7 at one’s place then its cube has 3 at one’s place.
As ?? ?? = ???? (?? ?? has 7 at its unit place)
?? ?? = ?????? (?? ?? has 3 at its unit place)
? If a number has 2 at one’s place then its cube has 8 at one’s place, also if a
number has 8 at one’s place then its cube has 2 at one’s place.
As ?? ?? = ?? (?? ?? has 8 at its unit place)
?? ?? = ?????? (?? ?? has 2 at its unit place)
? For two consecutive natural numbers p and q (p>q).
?? ?? - ?? ?? = ?????? + ??
e.g
?? ?? - ?? ?? = ?????? - ???? = ???? = ?? (?? × ?? ) + ??
Page 2
Cube and Cube Roots
Points to Remember
? When a given number is multiplied by itself three times, the new number
obtained is called cube of the given number and given number is called cube root
of the new number i.e. a number ‘m’ is the cube root of a number ‘n’ if ?? = ?? ??
or ?? = v ?? ?? . symbol v
?? represents cube root. E.g. ?? ?? = ?? ???? ?? = v ?? ??
? Numbers 1729, 4104, 13832 etc. are called Hardy-Ramanujan numbers. They
can be expressed as the sum of two cubes in two different ways.
???????? = ?? ?? + ????
?? ???? ?? ?? + ????
??
? Cube of a number having 1, 4, 6 or 9 as one’s digit has the same digit at one’s
place.
As ?? ?? = ?? (?? ?? has 1 at its unit place)
?? ?? = ???? (?? ?? has 4 at its unit place)
?? ?? = ?????? (?? ?? has 9 at its unit place)
? If a number has 3 at its one’s place, then its cube has 7 at its one’s place, also if a
number has 7 at one’s place then its cube has 3 at one’s place.
As ?? ?? = ???? (?? ?? has 7 at its unit place)
?? ?? = ?????? (?? ?? has 3 at its unit place)
? If a number has 2 at one’s place then its cube has 8 at one’s place, also if a
number has 8 at one’s place then its cube has 2 at one’s place.
As ?? ?? = ?? (?? ?? has 8 at its unit place)
?? ?? = ?????? (?? ?? has 2 at its unit place)
? For two consecutive natural numbers p and q (p>q).
?? ?? - ?? ?? = ?????? + ??
e.g
?? ?? - ?? ?? = ?????? - ???? = ???? = ?? (?? × ?? ) + ??
? For any positive integer ‘n’, we have
v -?? ?? = - v ?? ??
? For any two integers a and b we have
i. v????
?? = v ?? ?? × v?? ??
ii. v
?? ?? ?? =
v ?? ?? v?? ?? , ?? ? ??
QUESTIONS
1. If ?? = v
????
?? .????
?? , find ?? +
?? ?? .
2. Find the value of v
????????
????
?? .
3. Simplify:v?? ????
??????
?? .
4. Find the smallest natural number other than 1, which is a perfect square as well
as perfect cube.
5. Find the maximum number of digits in the cube of 4-digit number.
6. Find the minimum number of digit in the cube of a three-digit number.
7. If ?????? = ?? ?? then find the value of ?? ?? (?? ? ?? ).
8. Evaluate (?? . ?? )
?? - (?? . ?? )
?? .
9. Write the unit’s digit of v ?????? × ????????
??
10. If (?????? )
?? = ???? , then find x.
11. Evaluate (
?? ?? )
?? - (
?? ?? )
??
12. Simplify: v(?????? × ???? )
??
13. Simplify: v(-???? × ???? × ?? )
??
14. Simplify: ?? ????
+ ?? ????
+ ?? ?? ??
15. Simplify: ?? ????
× ?? -????
× ?? ??
16. ?? ?? + ?? ?? + ?? ?? =
?? ??????
, then find x.
17. Evaluate v(?? × ?? × ???? × ?? × ?? × ?? × ?? )
??
18. Volume of a cube is 729 cm
3
. Then find the area of its face.
19. Evaluate v(
?? .????
???? .????
)
??
Page 3
Cube and Cube Roots
Points to Remember
? When a given number is multiplied by itself three times, the new number
obtained is called cube of the given number and given number is called cube root
of the new number i.e. a number ‘m’ is the cube root of a number ‘n’ if ?? = ?? ??
or ?? = v ?? ?? . symbol v
?? represents cube root. E.g. ?? ?? = ?? ???? ?? = v ?? ??
? Numbers 1729, 4104, 13832 etc. are called Hardy-Ramanujan numbers. They
can be expressed as the sum of two cubes in two different ways.
???????? = ?? ?? + ????
?? ???? ?? ?? + ????
??
? Cube of a number having 1, 4, 6 or 9 as one’s digit has the same digit at one’s
place.
As ?? ?? = ?? (?? ?? has 1 at its unit place)
?? ?? = ???? (?? ?? has 4 at its unit place)
?? ?? = ?????? (?? ?? has 9 at its unit place)
? If a number has 3 at its one’s place, then its cube has 7 at its one’s place, also if a
number has 7 at one’s place then its cube has 3 at one’s place.
As ?? ?? = ???? (?? ?? has 7 at its unit place)
?? ?? = ?????? (?? ?? has 3 at its unit place)
? If a number has 2 at one’s place then its cube has 8 at one’s place, also if a
number has 8 at one’s place then its cube has 2 at one’s place.
As ?? ?? = ?? (?? ?? has 8 at its unit place)
?? ?? = ?????? (?? ?? has 2 at its unit place)
? For two consecutive natural numbers p and q (p>q).
?? ?? - ?? ?? = ?????? + ??
e.g
?? ?? - ?? ?? = ?????? - ???? = ???? = ?? (?? × ?? ) + ??
? For any positive integer ‘n’, we have
v -?? ?? = - v ?? ??
? For any two integers a and b we have
i. v????
?? = v ?? ?? × v?? ??
ii. v
?? ?? ?? =
v ?? ?? v?? ?? , ?? ? ??
QUESTIONS
1. If ?? = v
????
?? .????
?? , find ?? +
?? ?? .
2. Find the value of v
????????
????
?? .
3. Simplify:v?? ????
??????
?? .
4. Find the smallest natural number other than 1, which is a perfect square as well
as perfect cube.
5. Find the maximum number of digits in the cube of 4-digit number.
6. Find the minimum number of digit in the cube of a three-digit number.
7. If ?????? = ?? ?? then find the value of ?? ?? (?? ? ?? ).
8. Evaluate (?? . ?? )
?? - (?? . ?? )
?? .
9. Write the unit’s digit of v ?????? × ????????
??
10. If (?????? )
?? = ???? , then find x.
11. Evaluate (
?? ?? )
?? - (
?? ?? )
??
12. Simplify: v(?????? × ???? )
??
13. Simplify: v(-???? × ???? × ?? )
??
14. Simplify: ?? ????
+ ?? ????
+ ?? ?? ??
15. Simplify: ?? ????
× ?? -????
× ?? ??
16. ?? ?? + ?? ?? + ?? ?? =
?? ??????
, then find x.
17. Evaluate v(?? × ?? × ???? × ?? × ?? × ?? × ?? )
??
18. Volume of a cube is 729 cm
3
. Then find the area of its face.
19. Evaluate v(
?? .????
???? .????
)
??
20. Evaluate {????
?? + (????
?? )
?? ?? }
??
21. Evaluate {(????
?? + ?? ?? )
?? ?? }
??
22. The figure shows a cuboid with volume 1800 cm
3
. Find the value of x.
23. Find the smallest number that can be expressed in two different ways as sum of
two different cubes. What is the special name given to such numbers?
24. Find the largest negative number which is a perfect cube.
25. If ?????? = ?? × ?? × ?? × ?? × ?? , 28=2× ?? × ?? , and ???? = ?? × ?? × ?? × ??
Find the least number by which ?????? × ???? × ???? should be multiplied to get a perfect
cube.
26. Area of one face of a cube is 121 cm
2
. Find the volume of the cube.
27. Find the value of ????
?? - ????
?? if ????
?? + ????
?? + ???? × ???? =1029.
28. Find the value of ????
?? + ????
?? if ????
?? + ????
?? - ???? × ???? =237.
29. Find the unit digit of (???????? )
??
30. Evaluate v
??????????
??????
?? ÷ v
?? ??????
??
31. Find the number of unit cubes in the given figure.
D
C
B
A
F
E
G (x+3) cm
H
6 cm
3 cm
Page 4
Cube and Cube Roots
Points to Remember
? When a given number is multiplied by itself three times, the new number
obtained is called cube of the given number and given number is called cube root
of the new number i.e. a number ‘m’ is the cube root of a number ‘n’ if ?? = ?? ??
or ?? = v ?? ?? . symbol v
?? represents cube root. E.g. ?? ?? = ?? ???? ?? = v ?? ??
? Numbers 1729, 4104, 13832 etc. are called Hardy-Ramanujan numbers. They
can be expressed as the sum of two cubes in two different ways.
???????? = ?? ?? + ????
?? ???? ?? ?? + ????
??
? Cube of a number having 1, 4, 6 or 9 as one’s digit has the same digit at one’s
place.
As ?? ?? = ?? (?? ?? has 1 at its unit place)
?? ?? = ???? (?? ?? has 4 at its unit place)
?? ?? = ?????? (?? ?? has 9 at its unit place)
? If a number has 3 at its one’s place, then its cube has 7 at its one’s place, also if a
number has 7 at one’s place then its cube has 3 at one’s place.
As ?? ?? = ???? (?? ?? has 7 at its unit place)
?? ?? = ?????? (?? ?? has 3 at its unit place)
? If a number has 2 at one’s place then its cube has 8 at one’s place, also if a
number has 8 at one’s place then its cube has 2 at one’s place.
As ?? ?? = ?? (?? ?? has 8 at its unit place)
?? ?? = ?????? (?? ?? has 2 at its unit place)
? For two consecutive natural numbers p and q (p>q).
?? ?? - ?? ?? = ?????? + ??
e.g
?? ?? - ?? ?? = ?????? - ???? = ???? = ?? (?? × ?? ) + ??
? For any positive integer ‘n’, we have
v -?? ?? = - v ?? ??
? For any two integers a and b we have
i. v????
?? = v ?? ?? × v?? ??
ii. v
?? ?? ?? =
v ?? ?? v?? ?? , ?? ? ??
QUESTIONS
1. If ?? = v
????
?? .????
?? , find ?? +
?? ?? .
2. Find the value of v
????????
????
?? .
3. Simplify:v?? ????
??????
?? .
4. Find the smallest natural number other than 1, which is a perfect square as well
as perfect cube.
5. Find the maximum number of digits in the cube of 4-digit number.
6. Find the minimum number of digit in the cube of a three-digit number.
7. If ?????? = ?? ?? then find the value of ?? ?? (?? ? ?? ).
8. Evaluate (?? . ?? )
?? - (?? . ?? )
?? .
9. Write the unit’s digit of v ?????? × ????????
??
10. If (?????? )
?? = ???? , then find x.
11. Evaluate (
?? ?? )
?? - (
?? ?? )
??
12. Simplify: v(?????? × ???? )
??
13. Simplify: v(-???? × ???? × ?? )
??
14. Simplify: ?? ????
+ ?? ????
+ ?? ?? ??
15. Simplify: ?? ????
× ?? -????
× ?? ??
16. ?? ?? + ?? ?? + ?? ?? =
?? ??????
, then find x.
17. Evaluate v(?? × ?? × ???? × ?? × ?? × ?? × ?? )
??
18. Volume of a cube is 729 cm
3
. Then find the area of its face.
19. Evaluate v(
?? .????
???? .????
)
??
20. Evaluate {????
?? + (????
?? )
?? ?? }
??
21. Evaluate {(????
?? + ?? ?? )
?? ?? }
??
22. The figure shows a cuboid with volume 1800 cm
3
. Find the value of x.
23. Find the smallest number that can be expressed in two different ways as sum of
two different cubes. What is the special name given to such numbers?
24. Find the largest negative number which is a perfect cube.
25. If ?????? = ?? × ?? × ?? × ?? × ?? , 28=2× ?? × ?? , and ???? = ?? × ?? × ?? × ??
Find the least number by which ?????? × ???? × ???? should be multiplied to get a perfect
cube.
26. Area of one face of a cube is 121 cm
2
. Find the volume of the cube.
27. Find the value of ????
?? - ????
?? if ????
?? + ????
?? + ???? × ???? =1029.
28. Find the value of ????
?? + ????
?? if ????
?? + ????
?? - ???? × ???? =237.
29. Find the unit digit of (???????? )
??
30. Evaluate v
??????????
??????
?? ÷ v
?? ??????
??
31. Find the number of unit cubes in the given figure.
D
C
B
A
F
E
G (x+3) cm
H
6 cm
3 cm
32. How many cubes of edges 4 cm will be obtained on melting a solid of edge 12 cm?
33. Find the volume of a cube of edge 12cm.
34. Find the cube root of 46656.
35. The volume of a cube is 9261000 m
3
. Find the edge of the cube.
36. Find the cube root of (-1728)×125.
37. Find the number of thousand in ???? × ?? ?? × ?? ?? .
38. Find the number of hundreds in ???? × ?? ?? × ?? ?? .
39. Two cubes have their volumes in the ratio 1:8. If the volume of smaller cube is
125 cubic cm. Find the side of another cube.
40. Simplify v?? ??????
??????
??
41. Evaluate ????
?? - ?? ??
42. Evaluate
v
?????????? +
v
?? . ?? + v ?? . ??????
?? ?? ??
43. How many consecutive odd numbers will be needed to obtain the sum as ?? ?? .
44. Find x if ?? ???? -?? = ?????? .
45. Evaluate :[(????
?? + ????
?? )
?? ?? ]
??
Page 5
Cube and Cube Roots
Points to Remember
? When a given number is multiplied by itself three times, the new number
obtained is called cube of the given number and given number is called cube root
of the new number i.e. a number ‘m’ is the cube root of a number ‘n’ if ?? = ?? ??
or ?? = v ?? ?? . symbol v
?? represents cube root. E.g. ?? ?? = ?? ???? ?? = v ?? ??
? Numbers 1729, 4104, 13832 etc. are called Hardy-Ramanujan numbers. They
can be expressed as the sum of two cubes in two different ways.
???????? = ?? ?? + ????
?? ???? ?? ?? + ????
??
? Cube of a number having 1, 4, 6 or 9 as one’s digit has the same digit at one’s
place.
As ?? ?? = ?? (?? ?? has 1 at its unit place)
?? ?? = ???? (?? ?? has 4 at its unit place)
?? ?? = ?????? (?? ?? has 9 at its unit place)
? If a number has 3 at its one’s place, then its cube has 7 at its one’s place, also if a
number has 7 at one’s place then its cube has 3 at one’s place.
As ?? ?? = ???? (?? ?? has 7 at its unit place)
?? ?? = ?????? (?? ?? has 3 at its unit place)
? If a number has 2 at one’s place then its cube has 8 at one’s place, also if a
number has 8 at one’s place then its cube has 2 at one’s place.
As ?? ?? = ?? (?? ?? has 8 at its unit place)
?? ?? = ?????? (?? ?? has 2 at its unit place)
? For two consecutive natural numbers p and q (p>q).
?? ?? - ?? ?? = ?????? + ??
e.g
?? ?? - ?? ?? = ?????? - ???? = ???? = ?? (?? × ?? ) + ??
? For any positive integer ‘n’, we have
v -?? ?? = - v ?? ??
? For any two integers a and b we have
i. v????
?? = v ?? ?? × v?? ??
ii. v
?? ?? ?? =
v ?? ?? v?? ?? , ?? ? ??
QUESTIONS
1. If ?? = v
????
?? .????
?? , find ?? +
?? ?? .
2. Find the value of v
????????
????
?? .
3. Simplify:v?? ????
??????
?? .
4. Find the smallest natural number other than 1, which is a perfect square as well
as perfect cube.
5. Find the maximum number of digits in the cube of 4-digit number.
6. Find the minimum number of digit in the cube of a three-digit number.
7. If ?????? = ?? ?? then find the value of ?? ?? (?? ? ?? ).
8. Evaluate (?? . ?? )
?? - (?? . ?? )
?? .
9. Write the unit’s digit of v ?????? × ????????
??
10. If (?????? )
?? = ???? , then find x.
11. Evaluate (
?? ?? )
?? - (
?? ?? )
??
12. Simplify: v(?????? × ???? )
??
13. Simplify: v(-???? × ???? × ?? )
??
14. Simplify: ?? ????
+ ?? ????
+ ?? ?? ??
15. Simplify: ?? ????
× ?? -????
× ?? ??
16. ?? ?? + ?? ?? + ?? ?? =
?? ??????
, then find x.
17. Evaluate v(?? × ?? × ???? × ?? × ?? × ?? × ?? )
??
18. Volume of a cube is 729 cm
3
. Then find the area of its face.
19. Evaluate v(
?? .????
???? .????
)
??
20. Evaluate {????
?? + (????
?? )
?? ?? }
??
21. Evaluate {(????
?? + ?? ?? )
?? ?? }
??
22. The figure shows a cuboid with volume 1800 cm
3
. Find the value of x.
23. Find the smallest number that can be expressed in two different ways as sum of
two different cubes. What is the special name given to such numbers?
24. Find the largest negative number which is a perfect cube.
25. If ?????? = ?? × ?? × ?? × ?? × ?? , 28=2× ?? × ?? , and ???? = ?? × ?? × ?? × ??
Find the least number by which ?????? × ???? × ???? should be multiplied to get a perfect
cube.
26. Area of one face of a cube is 121 cm
2
. Find the volume of the cube.
27. Find the value of ????
?? - ????
?? if ????
?? + ????
?? + ???? × ???? =1029.
28. Find the value of ????
?? + ????
?? if ????
?? + ????
?? - ???? × ???? =237.
29. Find the unit digit of (???????? )
??
30. Evaluate v
??????????
??????
?? ÷ v
?? ??????
??
31. Find the number of unit cubes in the given figure.
D
C
B
A
F
E
G (x+3) cm
H
6 cm
3 cm
32. How many cubes of edges 4 cm will be obtained on melting a solid of edge 12 cm?
33. Find the volume of a cube of edge 12cm.
34. Find the cube root of 46656.
35. The volume of a cube is 9261000 m
3
. Find the edge of the cube.
36. Find the cube root of (-1728)×125.
37. Find the number of thousand in ???? × ?? ?? × ?? ?? .
38. Find the number of hundreds in ???? × ?? ?? × ?? ?? .
39. Two cubes have their volumes in the ratio 1:8. If the volume of smaller cube is
125 cubic cm. Find the side of another cube.
40. Simplify v?? ??????
??????
??
41. Evaluate ????
?? - ?? ??
42. Evaluate
v
?????????? +
v
?? . ?? + v ?? . ??????
?? ?? ??
43. How many consecutive odd numbers will be needed to obtain the sum as ?? ?? .
44. Find x if ?? ???? -?? = ?????? .
45. Evaluate :[(????
?? + ????
?? )
?? ?? ]
??
ANSWERS
Question No. Answer Question No. Answer
1. 4 23.
1729, Ramanujan
Hardy Numbers
2.
?? ?? ??
24. -1
3. 1
?? ?? 25. 18
4. 64 26. 1331 cubic cm
5. 12 27. 3087
6. 7 28. 7110
7. 16 29. 3
8. 0.019 30. 20
9. 2 31. 60
10.
?? ?? 32. 27
11.
????
??????
33. 1728 cubic cm
12. 28 34. 36
13. -20 35. 210m
14. ?? ????
36. -60
15. 49 37. 240 thousand
16. -3 38. 56 hundred
17. 42 39. 10 cm
18. 81 square cm 40.
?? ?? ??
19.
?? ????
41. 271
20. 1331 42. 25
21. 15625 43. 5
22. 97 cm 44. 2
45. 8000
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Dec 23, 2024 Last updated |
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The content of Mental Maths Questions and Answers: Cube and Cube Roots is prepared as per the latest Class 8 syllabus.
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