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Page 1
Square and Square Roots
Points to Remember
? A Natural number x is a perfect square if there exists a natural number y such
that x=y
2
.
? A number ending in 2, 3, 7 or 8 is never a perfect square.
? The number of zeros in the end of a perfect square is never odd.
? Square of even numbers are always even.
? Square of odd numbers are always odd.
? Sum of first ?? natural numbers=
?? (?? +?? )
??
? Sum of first ?? natural even numbers=?? (?? + ?? )
? Sum of first ?? natural odd numbers=?? ??
? The square of a natural number other than 1 is either a multiple of 4 or exceeds
a multiple of 4 by 1.
? The square of a natural number other than 1 is either a multiple of 3 or exceeds
a multiple of 3 by 1.
? For any number ?? greater than 1, (?? ?? , ?? ?? - ?? , ?? ?? + ?? )is a Pythagorean triplet.
QUESTIONS
1. How many perfect squares lie between 100 and 1000?
2. Write the smallest four digit perfect square number.
3. Find the value of x, if (?? )
?? + (?? )
?? + (???? )
?? = (?? )
??
4. Simplify:v ???? × ?? + ???? × ??
5. Simplify:
v
(?? . ???? ) + v ?? . ????????
6. Simplify:
v
?????? +
v
?? + v ????
7. Simplify:v???????? × v
????????
?? .????
8. Find the value of (?????? )
?? - (?????? )
??
9. Simplify: ?? ?? - ?? ?? + ?? ?? - ?? ?? + ?? ??
Page 2
Square and Square Roots
Points to Remember
? A Natural number x is a perfect square if there exists a natural number y such
that x=y
2
.
? A number ending in 2, 3, 7 or 8 is never a perfect square.
? The number of zeros in the end of a perfect square is never odd.
? Square of even numbers are always even.
? Square of odd numbers are always odd.
? Sum of first ?? natural numbers=
?? (?? +?? )
??
? Sum of first ?? natural even numbers=?? (?? + ?? )
? Sum of first ?? natural odd numbers=?? ??
? The square of a natural number other than 1 is either a multiple of 4 or exceeds
a multiple of 4 by 1.
? The square of a natural number other than 1 is either a multiple of 3 or exceeds
a multiple of 3 by 1.
? For any number ?? greater than 1, (?? ?? , ?? ?? - ?? , ?? ?? + ?? )is a Pythagorean triplet.
QUESTIONS
1. How many perfect squares lie between 100 and 1000?
2. Write the smallest four digit perfect square number.
3. Find the value of x, if (?? )
?? + (?? )
?? + (???? )
?? = (?? )
??
4. Simplify:v ???? × ?? + ???? × ??
5. Simplify:
v
(?? . ???? ) + v ?? . ????????
6. Simplify:
v
?????? +
v
?? + v ????
7. Simplify:v???????? × v
????????
?? .????
8. Find the value of (?????? )
?? - (?????? )
??
9. Simplify: ?? ?? - ?? ?? + ?? ?? - ?? ?? + ?? ??
10. Simplify:(v?????? ÷ ?? )
??
11. Simplify:v?????? + v ?? . ????
12. Simplify:v
?? ???? .??
13. How many 2’s are there in the prime factorization of 4000?
14. Find the least number that should be added to 221 to get a perfect square.
15. If area of a square field is 1764 sq. m. Find its perimeter.
16. How many numbers lie between (???? )
?? and (???? )
?? ?
17. Find the value of(???? + ?? + ?? + ?? + ?? + ?? )
?? .
18. Simplify: (v ???????? - v ?????? )
??
19. How many digits are there in the square root of 15625?
20. How many digits will be there in the square of 999?
21. Find the value of
v
???? +
v
???? +
v
?? ?? + v ??
22. Find the value of
v
???? -
v
???? +
v
???? - v ??
23. Find the value of ?? + ?? + ?? + ?? + ???? + ???? + ???? + ???? + ???? + ???? + ????
24. Find the value of x if, (?? )
?? + (?? )
?? + (???? )
?? = (?? )
??
25. Find the value of ?? + ?? + ?? + ?? + ???? + ???? + ???? + ???? + ???? + ????
26. Find the value of x if, (?? )
?? + (?? )
?? + (???? )
?? = (?? )
??
27. Find the value of (?? × ?? ) + (?? × ?? ) + (?? × ?? ) + (?? × ?? )
28. Find the value of (???? )
?? - (???? )
?? .
29. Find the value of v???? × v ????
30. What is the sum of first 14 odd natural numbers?
31. Find X and Y if (?? )
?? = ?????? such that ?? > ?? , where X and Y are positive
integers.
32. If length of a rectangular park is 80 m and its breadth is 60 m. find the sum of
length of its diagonals.
33. Evaluate: (?? + v ?? )
?? + (?? - v ?? )
?? .
34. Find the least number that should be subtracted from 537 to get a perfect
square.
35. If the area of a square is 20.25 sq. m, find its perimeter.
Page 3
Square and Square Roots
Points to Remember
? A Natural number x is a perfect square if there exists a natural number y such
that x=y
2
.
? A number ending in 2, 3, 7 or 8 is never a perfect square.
? The number of zeros in the end of a perfect square is never odd.
? Square of even numbers are always even.
? Square of odd numbers are always odd.
? Sum of first ?? natural numbers=
?? (?? +?? )
??
? Sum of first ?? natural even numbers=?? (?? + ?? )
? Sum of first ?? natural odd numbers=?? ??
? The square of a natural number other than 1 is either a multiple of 4 or exceeds
a multiple of 4 by 1.
? The square of a natural number other than 1 is either a multiple of 3 or exceeds
a multiple of 3 by 1.
? For any number ?? greater than 1, (?? ?? , ?? ?? - ?? , ?? ?? + ?? )is a Pythagorean triplet.
QUESTIONS
1. How many perfect squares lie between 100 and 1000?
2. Write the smallest four digit perfect square number.
3. Find the value of x, if (?? )
?? + (?? )
?? + (???? )
?? = (?? )
??
4. Simplify:v ???? × ?? + ???? × ??
5. Simplify:
v
(?? . ???? ) + v ?? . ????????
6. Simplify:
v
?????? +
v
?? + v ????
7. Simplify:v???????? × v
????????
?? .????
8. Find the value of (?????? )
?? - (?????? )
??
9. Simplify: ?? ?? - ?? ?? + ?? ?? - ?? ?? + ?? ??
10. Simplify:(v?????? ÷ ?? )
??
11. Simplify:v?????? + v ?? . ????
12. Simplify:v
?? ???? .??
13. How many 2’s are there in the prime factorization of 4000?
14. Find the least number that should be added to 221 to get a perfect square.
15. If area of a square field is 1764 sq. m. Find its perimeter.
16. How many numbers lie between (???? )
?? and (???? )
?? ?
17. Find the value of(???? + ?? + ?? + ?? + ?? + ?? )
?? .
18. Simplify: (v ???????? - v ?????? )
??
19. How many digits are there in the square root of 15625?
20. How many digits will be there in the square of 999?
21. Find the value of
v
???? +
v
???? +
v
?? ?? + v ??
22. Find the value of
v
???? -
v
???? +
v
???? - v ??
23. Find the value of ?? + ?? + ?? + ?? + ???? + ???? + ???? + ???? + ???? + ???? + ????
24. Find the value of x if, (?? )
?? + (?? )
?? + (???? )
?? = (?? )
??
25. Find the value of ?? + ?? + ?? + ?? + ???? + ???? + ???? + ???? + ???? + ????
26. Find the value of x if, (?? )
?? + (?? )
?? + (???? )
?? = (?? )
??
27. Find the value of (?? × ?? ) + (?? × ?? ) + (?? × ?? ) + (?? × ?? )
28. Find the value of (???? )
?? - (???? )
?? .
29. Find the value of v???? × v ????
30. What is the sum of first 14 odd natural numbers?
31. Find X and Y if (?? )
?? = ?????? such that ?? > ?? , where X and Y are positive
integers.
32. If length of a rectangular park is 80 m and its breadth is 60 m. find the sum of
length of its diagonals.
33. Evaluate: (?? + v ?? )
?? + (?? - v ?? )
?? .
34. Find the least number that should be subtracted from 537 to get a perfect
square.
35. If the area of a square is 20.25 sq. m, find its perimeter.
36. Find the least number by which 288 must be multiplied so that it becomes a
perfect square.
37. Find the least number by which 147 must be divided so that it becomes a perfect
square.
38. The area of a square field is ????
?? ?? sq. m. Find the length of the side of the square.
39. Find the length of a side of a square playground whose area is equal to the area
of rectangular field of dimension 72m and 50m.
40. Find the value of
v?????? -v ??????
v?????? +v ??????
41. Find: 9+11+13+15+17+19+21+23
42. How many perfect squares lie between 0 and 550?
43. Find the value of
v
v
????????
????????
44. Find the value of (???? + ?? )
?? - (???? - ?? )
??
45. Find the value of (v ?????? - v ?????? + v?????? - v ???? )
46. Find the missing number: ?????? × ___ = (???? )
??
47. Find the Pythagorean triplet whose smallest number is 12.
48. Find ?? if, ?? ?? = (???? )
?? - (???? )
??
49. Find ?? if, ?? = v ?? . ???? + ?? . ???? + ?? . ???? + ?? . ???? + ?? . ????
50. Find ?? if, ?? = (?? + ?? v?? )(?? - ?? v?? )
Page 4
Square and Square Roots
Points to Remember
? A Natural number x is a perfect square if there exists a natural number y such
that x=y
2
.
? A number ending in 2, 3, 7 or 8 is never a perfect square.
? The number of zeros in the end of a perfect square is never odd.
? Square of even numbers are always even.
? Square of odd numbers are always odd.
? Sum of first ?? natural numbers=
?? (?? +?? )
??
? Sum of first ?? natural even numbers=?? (?? + ?? )
? Sum of first ?? natural odd numbers=?? ??
? The square of a natural number other than 1 is either a multiple of 4 or exceeds
a multiple of 4 by 1.
? The square of a natural number other than 1 is either a multiple of 3 or exceeds
a multiple of 3 by 1.
? For any number ?? greater than 1, (?? ?? , ?? ?? - ?? , ?? ?? + ?? )is a Pythagorean triplet.
QUESTIONS
1. How many perfect squares lie between 100 and 1000?
2. Write the smallest four digit perfect square number.
3. Find the value of x, if (?? )
?? + (?? )
?? + (???? )
?? = (?? )
??
4. Simplify:v ???? × ?? + ???? × ??
5. Simplify:
v
(?? . ???? ) + v ?? . ????????
6. Simplify:
v
?????? +
v
?? + v ????
7. Simplify:v???????? × v
????????
?? .????
8. Find the value of (?????? )
?? - (?????? )
??
9. Simplify: ?? ?? - ?? ?? + ?? ?? - ?? ?? + ?? ??
10. Simplify:(v?????? ÷ ?? )
??
11. Simplify:v?????? + v ?? . ????
12. Simplify:v
?? ???? .??
13. How many 2’s are there in the prime factorization of 4000?
14. Find the least number that should be added to 221 to get a perfect square.
15. If area of a square field is 1764 sq. m. Find its perimeter.
16. How many numbers lie between (???? )
?? and (???? )
?? ?
17. Find the value of(???? + ?? + ?? + ?? + ?? + ?? )
?? .
18. Simplify: (v ???????? - v ?????? )
??
19. How many digits are there in the square root of 15625?
20. How many digits will be there in the square of 999?
21. Find the value of
v
???? +
v
???? +
v
?? ?? + v ??
22. Find the value of
v
???? -
v
???? +
v
???? - v ??
23. Find the value of ?? + ?? + ?? + ?? + ???? + ???? + ???? + ???? + ???? + ???? + ????
24. Find the value of x if, (?? )
?? + (?? )
?? + (???? )
?? = (?? )
??
25. Find the value of ?? + ?? + ?? + ?? + ???? + ???? + ???? + ???? + ???? + ????
26. Find the value of x if, (?? )
?? + (?? )
?? + (???? )
?? = (?? )
??
27. Find the value of (?? × ?? ) + (?? × ?? ) + (?? × ?? ) + (?? × ?? )
28. Find the value of (???? )
?? - (???? )
?? .
29. Find the value of v???? × v ????
30. What is the sum of first 14 odd natural numbers?
31. Find X and Y if (?? )
?? = ?????? such that ?? > ?? , where X and Y are positive
integers.
32. If length of a rectangular park is 80 m and its breadth is 60 m. find the sum of
length of its diagonals.
33. Evaluate: (?? + v ?? )
?? + (?? - v ?? )
?? .
34. Find the least number that should be subtracted from 537 to get a perfect
square.
35. If the area of a square is 20.25 sq. m, find its perimeter.
36. Find the least number by which 288 must be multiplied so that it becomes a
perfect square.
37. Find the least number by which 147 must be divided so that it becomes a perfect
square.
38. The area of a square field is ????
?? ?? sq. m. Find the length of the side of the square.
39. Find the length of a side of a square playground whose area is equal to the area
of rectangular field of dimension 72m and 50m.
40. Find the value of
v?????? -v ??????
v?????? +v ??????
41. Find: 9+11+13+15+17+19+21+23
42. How many perfect squares lie between 0 and 550?
43. Find the value of
v
v
????????
????????
44. Find the value of (???? + ?? )
?? - (???? - ?? )
??
45. Find the value of (v ?????? - v ?????? + v?????? - v ???? )
46. Find the missing number: ?????? × ___ = (???? )
??
47. Find the Pythagorean triplet whose smallest number is 12.
48. Find ?? if, ?? ?? = (???? )
?? - (???? )
??
49. Find ?? if, ?? = v ?? . ???? + ?? . ???? + ?? . ???? + ?? . ???? + ?? . ????
50. Find ?? if, ?? = (?? + ?? v?? )(?? - ?? v?? )
ANSWERS
Question No. Answer Question No. Answer
1. 21 26. ?? = ????
2. 1024 27. 40
3. ?? = ???? 28. 120
4. 12 29. 66
5. 0.3 30. 196
6. 18 31. X= ???? , Y= ??
7. 1000 32. 200 m
8. 3003 33. 14
9. 15 34. 8
10. 9 35. 18 m
11. 10.3 36. 2
12.
?? ????
37. 3
13. 5 38. 5.5 m
14. 4 39. 60 m
15. 168 m 40.
?? ????
16. 124 41. 128
17. 625 42. 23
18. 1 43.
?? ??
19. 3 44. 528
20. 6 45. 10
21. 6 46. 7
22. 7 47. 12,16,20 or 12,35,37
23. 143 48. 93
24. ?? = ???? 49. 0.6
25. 110 50 1
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