NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots
NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots Exercise 6.1
Ex 6.1 Class 8 Maths Question 1.
What will be the unit digit of the squares of the following numbers?
(i) 81
(ii) 272
(iii) 799
(iv) 3853
(v) 1234
(vi) 20387
(vii) 52698
(viii) 99880
(ix) 12796
(x) 55555
Solution:
(i) Unit digit of 812Β = 1
(ii) Unit digit of 2722Β = 4
(iii) Unit digit of 7992Β = 1
(iv) Unit digit of 38532Β = 9
(v) Unit digit of 12342Β = 6
(vi) Unit digit of 263872Β = 9
(vii) Unit digit of 526982Β = 4
(viii) Unit digit of 998802Β = 0
(ix) Unit digit of 127962Β = 6
(x) Unit digit of 555552Β = 5
Ex 6.1 Class 8 MathsΒ Question 2.
The following numbers are not perfect squares. Give reason.
(i) 1057
(ii) 23453
(iii) 7928
(iv) 222222
(v) 64000
(vi) 89722
(vii) 222000
(viii) 505050
Solution:
(i) 1057 ends with 7 at unit place. So it is not a perfect square number.
(ii) 23453 ends with 3 at unit place. So it is not a perfect square number.
(iii) 7928 ends with 8 at unit place. So it is not a perfect square number.
(iv) 222222 ends with 2 at unit place. So it is not a perfect square number.
(v) 64000 ends with 3 zeros. So it cannot a perfect square number.
(vi) 89722 ends with 2 at unit place. So it is not a perfect square number.
(vii) 22000 ends with 3 zeros. So it can not be a perfect square number.
(viii) 505050 ends with 1 zero. So it is not a perfect square number.
Ex 6.1 Class 8 MathsΒ Question 3.
The squares of which of the following would be odd numbers?
(i) 431
(ii) 2826
(iii) 7779
(iv) 82004
Solution:
(i) 4312Β is an odd number.
(ii) 28262Β is an even number.
(iii) 77792Β is an odd number.
(iv) 820042Β is an even number.
Ex 6.1 Class 8 MathsΒ Question 4.
Observe the following pattern and find the missing digits.
112Β = 121
1012Β = 10201
10012Β = 1002001
1000012Β = 1β¦2β¦1
100000012Β = β¦β¦β¦
Solution:
According to the above pattern, we have
1000012Β = 10000200001
100000012Β = 100000020000001
Ex 6.1 Class 8 MathsΒ Question 5.
Observe the following pattern and supply the missing numbers.
112Β = 121
1012Β = 10201
101012Β = 102030201
10101012Β = β¦β¦β¦.
β¦β¦β¦.2Β = 10203040504030201
Solution:
According to the above pattern, we have
10101012Β = 1020304030201
1010101012Β = 10203040504030201
Ex 6.1 Class 8 MathsΒ Question 6.
Using the given pattern, find the missing numbers.
12Β + 22Β + 22Β = 32
22Β + 32Β + 62Β = 72
32Β + 42Β + 122Β = 132
42Β + 52Β + β¦.2Β = 212
52Β + β¦.2Β + 302Β = 312
62Β + 72Β + β¦..2Β = β¦β¦2
Solution:
According to the given pattern, we have
42Β + 52Β + 202Β = 212
52Β + 62Β + 302Β = 312
62Β + 72Β + 422Β = 432
Ex 6.1 Class 8 MathsΒ Question 7.
Without adding, find the sum.
(i) 1 + 3 + 5 + 7 + 9
(ii) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19
(iii) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23
Solution:
We know that the sum of n odd numbers = n2
(i) 1 + 3 + 5 + 7 + 9 = (5)2Β = 25 [β΅ n = 5]
(ii) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = (10)2Β = 100 [β΅ n = 10]
(iii) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 = (12)2Β = 144 [β΅ n = 12]
Ex 6.1 Class 8 MathsΒ Question 8.
(i) Express 49 as the sum of 7 odd numbers.
(ii) Express 121 as the sum of 11 odd numbers.
Solution:
(i) 49 = 1 + 3 + 5 + 7 + 9 + 11 + 13 (n = 7)
(ii) 121 = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 (n = 11)
Ex 6.1 Class 8 MathsΒ Question 9.
How many numbers lie between squares of the following numbers?
(i) 12 and 13
(ii) 25 and 26
(iii) 99 and 100.
Solution:
(i) We know that numbers between n2Β and (n + 1)2Β = 2n
Numbers between 122Β and 132Β = (2n) = 2 Γ 12 = 24
(ii) Numbers between 252Β and 262Β = 2 Γ 25 = 50 (β΅ n = 25)
(iii) Numbers between 992Β and 1002Β = 2 Γ 99 = 198 (β΅ n = 99)
NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots Exercise 6.2
Ex 6.2 Class 8 Maths Question 1.
Find the square of the following numbers.
(i) 32
(ii) 35
(iii) 86
(iv) 93
(v) 71
(vi) 46
Solution:
(i) 32 = 30 + 2
(32)2Β = (30 + 2)2
= 30(30 + 2) + 2(30 + 2)
= 302Β + 30 Γ 2 + 2 Γ 30 + 22
= 900 + 60 + 60 + 4
= 1024
Thus (32)2Β = 1024
(ii) 35 = (30 + 5)
(35)2Β = (30 + 5)2
= 30(30 + 5) + 5(30 + 5)
= (30)2Β + 30 Γ 5 + 5 Γ 30 + (5)2
= 900 + 150 + 150 + 25
= 1225
Thus (35)2Β = 1225
(iii) 86 = (80 + 6)
862Β = (80 + 6)2
= 80(80 + 6) + 6(80 + 6)
= (80)2Β + 80 Γ 6 + 6 Γ 80 + (6)2
= 6400 + 480 + 480 + 36
= 7396
Thus (86)2Β = 7396
(iv) 93 = (90+ 3)
932Β = (90 + 3)2
= 90 (90 + 3) + 3(90 + 3)
= (90)2Β + 90 Γ 3 + 3 Γ 90 + (3)2
= 8100 + 270 + 270 + 9
= 8649
Thus (93)2Β = 8649
(v) 71 = (70 + 1)
712Β = (70 + 1)2
= 70 (70 + 1) + 1(70 + 1)
= (70)2Β + 70 Γ 1 + 1 Γ 70 + (1)2
= 4900 + 70 + 70 + 1
= 5041
Thus (71)2Β = 5041
(vi) 46 = (40+ 6)
462Β = (40 + 6)2
= 40 (40 + 6) + 6(40 + 6)
= (40)2Β + 40 Γ 6 + 6 Γ 40 + (6)2
= 1600 + 240 + 240 + 36
= 2116
Thus (46)2Β = 2116
Ex 6.2 Class 8 MathsΒ Question 2.
Write a Pythagorean triplet whose one member is
(i) 6
(ii) 14
(iii) 16
(iv) 18
Solution:
(i) Let m2Β β 1 = 6
[Triplets are in the form 2m, m2Β β 1, m2Β + 1]
m2Β = 6 + 1 = 7
So, the value of m will not be an integer.
Now, let us try for m2Β + 1 = 6
β m2Β = 6 β 1 = 5
Also, the value of m will not be an integer.
Now we let 2m = 6 β m = 3 which is an integer.
Other members are:
m2Β β 1 = 32Β β 1 = 8 and m2Β + 1 = 32Β + 1 = 10
Hence, the required triplets are 6, 8 and 10
(ii) Let m2Β β 1 = 14 β m2Β = 1 + 14 = 15
The value of m will not be an integer.
Now take 2m = 14 β m = 7 which is an integer.
The member of triplets are 2m = 2 Γ 7 = 14
m2Β β 1 = (7)2Β β 1 = 49 β 1 = 48
and m2Β + 1 = (7)2Β + 1 = 49 + 1 = 50
i.e., (14, 48, 50)
(iii) Let 2m = 16 m = 8
The required triplets are 2m = 2 Γ 8 = 16
m2Β β 1 = (8)2Β β 1 = 64 β 1 = 63
m2Β + 1 = (8)2Β + 1 = 64 + 1 = 65
i.e., (16, 63, 65)
(iv) Let 2m = 18 β m = 9
Required triplets are:
2m = 2 Γ 9 = 18
m2Β β 1 = (9)2Β β 1 = 81 β 1 = 80
and m2Β + 1 = (9)2Β + 1 = 81 + 1 = 82
i.e., (18, 80, 82)
NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots Exercise 6.3
Ex 6.3 Class 8 Maths Question 1.
What could be the possible βoneβsβ digits of the square root of each of the following numbers?
(i) 9801
(ii) 99856
(iii) 998001
(iv) 657666025
Solution:
(i) Oneβs digit in the square root of 9801 maybe 1 or 9.
(ii) Oneβs digit in the square root of 99856 maybe 4 or 6.
(iii) Oneβs digit in the square root of 998001 maybe 1 or 9.
(iv) Oneβs digit in the square root of 657666025 can be 5.
Ex 6.3 Class 8 MathsΒ Question 2.
Without doing any calculation, find the numbers which are surely not perfect squares.
(i) 153
(ii) 257
(iii) 408
(iv) 441
Solution:
We know that the numbers ending with 2, 3, 7 or 8 are not perfect squares.
(i) 153 is not a perfect square number. (ending with 3)
(ii) 257 is not a perfect square number. (ending with 7)
(iii) 408 is not a perfect square number. (ending with 8)
(iv) 441 is a perfect square number.
Ex 6.3 Class 8 MathsΒ Question 3.
Find the square roots of 100 and 169 by the method of repeated subtraction.
Solution:
Using the method of repeated subtraction of consecutive odd numbers, we have
(i) 100 β 1 = 99, 99 β 3 = 96, 96 β 5 = 91, 91 β 7 = 84, 84 β 9 = 75, 75 β 11 = 64, 64 β 13 = 51, 51 β 15 = 36, 36 β 17 = 19, 19 β 19 = 0
(Ten times repetition)
Thus β100 = 10
(ii) 169 β 1 = 168, 168 β 3 = 165, 165 β 5 = 160, 160 β 7 = 153, 153 β 9 = 144, 144 β 11 = 133, 133 β 13 = 120, 120 β 15 = 105, 105 β 17 = 88, 88 β 19 = 69, 69 β 21 = 48, 48 β 23 = 25, 25 β 25 = 0
(Thirteen times repetition)
Thus β169 = 13
Ex 6.3 Class 8 MathsΒ Question 4.
Find the square roots of the following numbers by the prime factorisation Method.
(i) 729
(ii) 400
(iii) 1764
(iv) 4096
(v) 7744
(vi) 9604
(vii) 5929
(viii) 9216
(ix) 529
(x) 8100
Solution:
(i) We have 729
Prime factors of 729
729 = 3 Γ 3 Γ 3 Γ 3 Γ 3 Γ 3 = 32Β Γ 32Β Γ 32
β729 = 3 Γ 3 Γ 3 = 27
(ii) We have 400
Prime factors of 400
400 = 2 Γ 2 Γ 2 Γ 2 Γ 5 Γ 5 = 22Β Γ 22Β Γ 52
β400 = 2 Γ 2 Γ 5 = 20
(iii) 1764
1764 = 2 Γ 2 Γ 3 Γ 3 Γ 7 Γ 7 = 22Β Γ 32Β Γ 72
β1764 = 2 Γ 3 Γ 7 = 42
(iv) 4096
4096 = 2 Γ 2 Γ 2 Γ 2 Γ 2 Γ 2 Γ 2 Γ 2 Γ 2 Γ 2 Γ 2 Γ 2
= 22Β Γ 22Β Γ 22Β Γ 22Β Γ 22Β Γ 22
β4096 = 2 Γ 2 Γ 2 Γ 2 Γ 2 Γ 2 = 64
(v) Prime factorisation of 7744 is
7744 = 2 Γ 2 Γ 2 Γ 2 Γ 2 Γ 2 Γ 11 Γ 11
= 22Β Γ 22Β Γ 22Β Γ 112
β7744 = 2 Γ 2 Γ 2 Γ 11 = 88
(vi) Prime factorisation of 9604 is
9604 = 2 Γ 2 Γ 7 Γ 7 Γ 7 Γ 7 = 22Β Γ 72Β Γ 72
β9604 = 2 Γ 7 Γ 7 = 98
(vii) Prime factorisation of 5929 is
5929 = 7 Γ 7 Γ 11 Γ 11 = 72Β Γ 112
β5929 = 7 Γ 11 = 77
(viii) Prime factorisation of 9216 is
9216 = 2 Γ 2 Γ 2 Γ 2 Γ 2 Γ 2 Γ2 Γ 2 Γ 2 Γ 2 Γ 3 Γ 3
= 22Β Γ 22Β Γ 22Β Γ 22Β Γ 22Β Γ 32
β9216 = 2 Γ 2 Γ 2 Γ 2 Γ 2 Γ 3 = 96
(ix) Prime factorisation of 529 is
529 = 23 Γ 23 = 232
β529 = 23
(x) Prime factorisation of 8100 is
8100 = 2 Γ 2 Γ 3 Γ 3 Γ 3 Γ 3 Γ 5 Γ 5 = 22Β Γ 32Β Γ 32Β Γ 52
β8100 = 2 Γ 3 Γ 3 Γ 5 = 90
Ex 6.3 Class 8 MathsΒ Question 5.
For each of the following numbers, find the smallest whole number by which it should be multiplied so as to get a perfect square number. Also, find the square root of the square number so obtained.
(i) 252
(ii) 180
(iii) 1008
(iv) 2028
(v) 1458
(vi) 768
Solution:
(i) Prime factorisation of 252 is
252 = 2 Γ 2 Γ 3 Γ 3 Γ 7
Here, the prime factorisation is not in pair. 7 has no pair.
Thus, 7 is the smallest whole number by which the given number is multiplied to get a perfect square number.
The new square number is 252 Γ 7 = 1764
Square root of 1764 is
β1764 = 2 Γ 3 Γ 7 = 42
(ii) Primp factorisation of 180 is
180 = 2 Γ 2 Γ 3 Γ 3 Γ 5
Here, 5 has no pair.
New square number = 180 Γ 5 = 900
The square root of 900 is
β900 = 2 Γ 3 Γ 5 = 30
Thus, 5 is the smallest whole number by which the given number is multiplied to get a square number.
(iii) Prime factorisation of 1008 is
1008 = 2 Γ 2 Γ 2 Γ 2 Γ 3 Γ 3 Γ 7
Here, 7 has no pair.
New square number = 1008 Γ 7 = 7056
Thus, 7 is the required number.
Square root of 7056 is
β7056 = 2 Γ 2 Γ 3 Γ 7 = 84
(iv) Prime factorisation of 2028 is
2028 = 2 Γ 2 Γ 3 Γ 13 Γ 13
Here, 3 is not in pair.
Thus, 3 is the required smallest whole number.
New square number = 2028 Γ 3 = 6084
Square root of 6084 is
β6084 = 2 Γ 13 Γ 3 = 78
(v) Prime factorisation of 1458 is
1458 = 2 Γ 3 Γ 3 Γ 3 Γ 3 Γ 3 Γ 3
Here, 2 is not in pair.
Thus, 2 is the required smallest whole number.
New square number = 1458 Γ 2 = 2916
Square root of 1458 is
β2916 = 3 Γ 3 Γ 3 Γ 2 = 54
(vi) Prime factorisation of 768 is
768 = 2 Γ 2 Γ 2 Γ 2 Γ 2 Γ 2 Γ 2 Γ 2 Γ 3
Here, 3 is not in pair.
Thus, 3 is the required whole number.
New square number = 768 Γ 3 = 2304
Square root of 2304 is
β2304 = 2 Γ 2 Γ 2 Γ 2 Γ 3 = 48
Ex 6.3 Class 8 MathsΒ Question 6.
For each of the following numbers, find the smallest whole number by which it should be divided so as to get a perfect square. Also, find the square root of the square number so obtained.
(i) 252
(ii) 2925
(iii) 396
(iv) 2645
(v) 2800
(vi) 1620
Solution:
(i) Prime factorisation of 252 is
252 = 2 Γ 2 Γ 3 Γ 3 Γ 7
Here 7 has no pair.
7 is the smallest whole number by which 252 is divided to get a square number.
New square number = 252 Γ· 7 = 36
Thus, β36 = 6
(ii) Prime factorisation of 2925 is
2925 = 3 Γ 3 Γ 5 Γ 5 Γ 13
Here, 13 has no pair.
13 is the smallest whole number by which 2925 is divided to get a square number.
New square number = 2925 Γ· 13 = 225
Thus β225 = 15
(iii) Prime factorisation of 396 is
396 = 2 Γ 2 Γ 3 Γ 3 Γ 11
Here 11 is not in pair.
11 is the required smallest whole number by which 396 is divided to get a square number.
New square number = 396 Γ· 11 = 36
Thus β36 = 6
(iv) Prime factorisation of 2645 is
2645 = 5 Γ 23 Γ 23
Here, 5 is not in pair.
5 is the required smallest whole number.
By which 2645 is multiplied to get a square number
New square number = 2645 Γ· 5 = 529
Thus, β529 = 23
(v) Prime factorisation of 2800 is
2800 = 2 Γ 2 Γ 2 Γ 2 Γ 5 Γ 5 Γ 7
Here, 7 is not in pair.
7 is the required smallest number.
By which 2800 is multiplied to get a square number.
New square number = 2800 Γ· 7 = 400
Thus β400 = 20
(vi) Prime factorisation of 1620 is
1620 = 2 Γ 2 Γ 3 Γ 3 Γ 3 Γ 3 Γ 5
Here, 5 is not in pair.
5 is the required smallest prime number.
By which 1620 is multiplied to get a square number = 1620 Γ· 5 = 324
Thus β324 = 18
Ex 6.3 Class 8 MathsΒ Question 7.
The students of class VIII of a school donated βΉ 2401 in all, for Prime Ministerβs National Relief Fund. Each student donated as many rupees as the number of students in the class. Find the number of students in the class.
Solution:
Total amount of money donated = βΉ 2401
Total number of students in the class = β2401
=Β 72Γ72βββββββ
=Β 7Γ7Γ7Γ7βββββββββββ
= 7 Γ 7
= 49
Ex 6.3 Class 8 MathsΒ Question 8.
2025 plants are to be planted in a garden in such a way that each row contains as many plants as the number of rows. Find the number of rows and the number of plants in each row.
Solution:
Total number of rows = Total number of plants in each row = β2025
=Β 3Γ3Γ3Γ3Γ5Γ5βββββββββββββββββ
=Β 32Γ32Γ52βββββββββββ
= 3 Γ 3 Γ 5
= 45
Thus the number of rows and plants = 45
Ex 6.3 Class 8 MathsΒ Question 9.
Find the smallest square number that is divisible by each of the numbers 4, 9 and 10.
Solution:
LCM of 4, 9, 10 = 180
The least number divisible by 4, 9 and 10 = 180
Now prime factorisation of 180 is
180 = 2 Γ 2 Γ 3 Γ 3 Γ 5
Here, 5 has no pair.
The required smallest square number = 180 Γ 5 = 900
Ex 6.3 Class 8 MathsΒ Question 10.
Find the smallest number that is divisible by each of the numbers 8, 15 and 20.
Solution:
The smallest number divisible by 8, 15 and 20 is equal to their LCM.
LCM = 2 Γ 2 Γ 2 Γ 3 Γ 5 = 120
Here, 2, 3 and 5 have no pair.
The required smallest square number = 120 Γ 2 Γ 3 Γ 5 = 120 Γ 30 = 3600
NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots Exercise 6.4
Ex 6.4 Class 8 Maths Question 1.
Find the square root of each of the following numbers by Long Division method.
(i) 2304
(ii) 4489
(iii) 3481
(iv) 529
(v) 3249
(vi) 1369
(vii) 5776
(viii) 7921
(ix) 576
(x) 1024
(xi) 3136
(xii) 900
Solution:
Ex 6.4 Class 8 MathsΒ Question 2.
Find the number of digits in the square root of each of the following numbers (without any calculation)
(i) 64
(ii) 144
(iii) 4489
(iv) 27225
(v) 390625
Solution:
We know that if n is number of digits in a square number then
Number of digits in the square root =Β n2Β if n is even andΒ n+12Β if n is odd.
(i) 64
Here n = 2 (even)
Number of digits in β64 =Β 22Β = 1
(ii) 144
Here n = 3 (odd)
Number of digits in square root =Β 3+12Β = 2
(iii) 4489
Here n = 4 (even)
Number of digits in square root =Β 42Β = 2
(iv) 27225
Here n = 5 (odd)
Number of digits in square root =Β 5+12Β = 3
(iv) 390625
Here n = 6 (even)
Number of digits in square root =Β 62Β = 3
Ex 6.4 Class 8 MathsΒ Question 3.
Find the square root of the following decimal numbers.
(i) 2.56
(ii) 7.29
(iii) 51.84
(iv) 42.25
(v) 31.36
Solution:
Ex 6.4 Class 8 MathsΒ Question 4.
Find the least number which must be subtracted from each of the following numbers so as to get a perfect square. Also find the square root of the perfect square so obtained.
(i) 402
(ii) 1989
(iii) 3250
(iv) 825
(v) 4000
Solution:
(i)
Here remainder is 2
2 is the least required number to be subtracted from 402 to get a perfect square
New number = 402 β 2 = 400
Thus, β400 = 20
(ii)
Here remainder is 53
53 is the least required number to be subtracted from 1989.
New number = 1989 β 53 = 1936
Thus, β1936 = 44
(iii)
Here remainder is 1
1 is the least required number to be subtracted from 3250 to get a perfect square.
New number = 3250 β 1 = 3249
Thus, β3249 = 57
(iv)
Here, the remainder is 41
41 is the least required number which can be subtracted from 825 to get a perfect square.
New number = 825 β 41 = 784
Thus, β784 = 28
(v)
Here, the remainder is 31
31 is the least required number which should be subtracted from 4000 to get a perfect square.
New number = 4000 β 31 = 3969
Thus, β3969 = 63
Ex 6.4 Class 8 MathsΒ Question 5.
Find the least number which must be added to each of the following numbers so as to get a perfect square. Also, find the square root of the perfect square so obtained.
(i) 525
(ii) 1750
(iii) 252
(iv) 1825
(v) 6412
Solution:
(i)
Here remainder is 41
It represents that square of 22 is less than 525.
Next number is 23 an 232Β = 529
Hence, the number to be added = 529 β 525 = 4
New number = 529
Thus, β529 = 23
(ii)
Here the remainder is 69
It represents that square of 41 is less than in 1750.
The next number is 42 and 422Β = 1764
Hence, number to be added to 1750 = 1764 β 1750 = 14
Require perfect square = 1764
β1764 = 42
(iii)
Here the remainder is 27.
It represents that a square of 15 is less than 252.
The next number is 16 and 162Β = 256
Hence, number to be added to 252 = 256 β 252 = 4
New number = 252 + 4 = 256
Required perfect square = 256
and β256 = 16
(iv)
The remainder is 61.
It represents that square of 42 is less than in 1825.
Next number is 43 and 432Β = 1849
Hence, number to be added to 1825 = 1849 β 1825 = 24
The required perfect square is 1848 and β1849 =43
(v)
Here, the remainder is 12.
It represents that a square of 80 is less than in 6412.
The next number is 81 and 812Β = 6561
Hence the number to be added = 6561 β 6412 = 149
The require perfect square is 6561 and β6561 = 81
Ex 6.4 Class 8 MathsΒ Question 6.
Find the length of the side of a square whose area = 441 m2
Solution:
Let the length of the side of the square be x m.
Area of the square = (side)2Β = x2Β m2
x2Β = 441 β x = β441 = 21
Thus, x = 21 m.
Hence the length of the side of square = 21 m.
Ex 6.4 Class 8 MathsΒ Question 7.
In a right triangle ABC, β B = 90Β°.
(a) If AB = 6 cm, BC = 8 cm, find AC
(b) If AC = 13 cm, BC = 5 cm, find AB
Solution:
(a) In right triangle ABC
AC2Β = AB2Β + BC2Β [By Pythagoras Theorem]
β AC2Β = (6)2Β + (8)2Β = 36 + 64 = 100
β AC = β100 = 10
Thus, AC = 10 cm.
(b) In right triangle ABC
AC2Β = AB2Β + BC2Β [By Pythagoras Theorem]
β (13)2Β = AB2Β + (5)2
β 169 = AB2Β + 25
β 169 β 25 = AB2
β 144 = AB2
AB = β144 = 12 cm
Thus, AB = 12 cm.
Ex 6.4 Class 8 MathsΒ Question 8.
A gardener has 1000 plants. He wants to plant these in such a way that the number of rows and the number of columns remain the same. Find the minimum number of plants he needs more for this.
Solution:
Let the number of rows be x.
And the number of columns also be x.
Total number of plants = x Γ x = x2
x2Β = 1000 β x = β1000
Here the remainder is 39
So the square of 31 is less than 1000.
Next number is 32 and 322Β = 1024
Hence the number to be added = 1024 β 1000 = 24
Thus the minimum number of plants required by him = 24.
Alternative method:
The minimum number of plants required by him = 24.
Ex 6.4 Class 8 MathsΒ Question 9.
There are 500 children in a school. For a P.T. drill, they have to stand in such a manner that the number of rows is equal to the number of columns. How many children would be left out in this arrangement?
Solution:
Let the number of children in a row be x. And also that of in a column be x.
Total number of students = x Γ x = x2
x2Β = 500 β x = β500
Here the remainder is 16
New Number 500 β 16 = 484
and, β484 = 22
Thus, 16 students will be left out in this arrangement.
Squares and Square Roots Class 8 Extra Questions Maths Chapter 6
Extra Questions for Class 8 Maths Chapter 6 Squares and Square Roots
Question 1.
Find the perfect square numbers between 40 and 50.
Solution:
Perfect square numbers between 40 and 50 = 49.
Question 2.
Which of the following 242, 492, 772, 1312Β or 1892Β end with digit 1?
Solution:
Only 492, 1312Β and 1892Β end with digit 1.
Question 3.
Find the value of each of the following without calculating squares.
(i) 272Β β 262
(ii) 1182Β β 1172
Solution:
(i) 272Β β 262Β = 27 + 26 = 53
(ii) 1182Β β 1172Β = 118 + 117 = 235
Question 4.
Write each of the following numbers as difference of the square of two consecutive natural numbers.
(i) 49
(ii) 75
(iii) 125
Solution:
(i) 49 = 2 Γ 24 + 1
49 = 252Β β 242
(ii) 75 = 2 Γ 37 + 1
75 = 382Β β 372
(iii) 125 = 2 Γ 62 + 1
125 = 632Β β 622
Question 5.
Write down the following as sum of odd numbers.
(i) 72
(ii) 92
Solution:
(i) 72Β = Sum of first 7 odd numbers = 1 + 3 + 5 + 7 + 9 + 11 + 13
(ii) 92Β = Sum of first 9 odd numbers = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17
Question 6.
Express the following as the sum of two consecutive integers.
(i) 152
(ii) 192
Solution:
Question 7.
Find the product of the following:
(i) 23 Γ 25
(ii) 41 Γ 43
Solution:
(i) 23 Γ 25 = (24 β 1) (24 + 1) = 242Β β 1 = 576 β 1 = 575
(ii) 41 Γ 43 = (42 β 1) (42 + 1) = 422Β β 1 = 1764 β 1 = 1763
Question 8.
Find the squares of:
(i)Β β37
(ii)Β β917
Solution:
Question 9.
Check whether (6, 8, 10) is a Pythagorean triplet.
Solution:
2m, m2Β β 1 and m2Β + 1 represent the Pythagorean triplet.
Let 2m = 6 β m = 3
m2Β β 1 = (3)2Β β 1 = 9 β 1 = 8
and m2Β + 1 = (3)2Β + 1 = 9 + 1 = 10
Hence (6, 8, 10) is a Pythagorean triplet.
Alternative Method:
(6)2Β + (8)2Β = 36 + 64 = 100 = (10)2
β (6, 8, 10) is a Pythagorean triplet.
Question 10.
Using property, find the value of the following:
(i) 192Β β 182
(ii) 232Β β 222
Solution:
(i) 192Β β 182Β = 19 + 18 = 37
(ii) 232Β β 222Β = 23 + 22 = 45
Squares and Square Roots Class 8 Extra Questions Short Answer Type
Question 11.
Using the prime factorisation method, find which of the following numbers are not perfect squares.
(i) 768
(ii) 1296
Solution:
768 =Β 2 Γ 2Β ΓΒ 2 Γ 2Β ΓΒ 2 Γ 2Β ΓΒ 2 Γ 2Β Γ 3
Here, 3 is not in pair.
768 is not a perfect square.
1296 =Β 2 Γ 2Β ΓΒ 2 Γ 2Β ΓΒ 3 Γ 3Β ΓΒ 3 Γ 3
Here, there is no number left to make a pair.
1296 is a perfect square.
Question 12.
Which of the following triplets are Pythagorean?
(i) (14, 48, 50)
(ii) (18, 79, 82)
Solution:
We know that 2m, m2Β β 1 and m2Β + 1 make Pythagorean triplets.
(i) For (14, 48, 50),
Put 2m =14 β m = 7
m2Β β 1 = (7)2Β β 1 = 49 β 1 = 48
m2Β + 1 = (7)2Β + 1 = 49 + 1 = 50
Hence (14, 48, 50) is a Pythagorean triplet.
(ii) For (18, 79, 82)
Put 2m = 18 β m = 9
m2Β β 1 = (9)2Β β 1 = 81 β 1 = 80
m2Β + 1 = (9)2Β + 1 = 81 + 1 = 82
Hence (18, 79, 82) is not a Pythagorean triplet.
Question 13.
Find the square root of the following using successive subtraction of odd numbers starting from 1.
(i) 169
(ii) 81
(iii) 225
Solution:
(i) 169 β 1 = 168, 168 β 3 = 165, 165 β 5 = 160, 160 β 7 = 153, 153 β 9 = 144, 144 β 11 = 133, 133 β 13 = 120, 120 β 15 = 105, 105 β 17 = 88, 88 β 19 = 69,
69 β 21 = 48, 48 β 23 = 25, 25 β 25 = 0
We have subtracted odd numbers 13 times to get 0.
β169 = 13
(ii) 81 β 1 = 80, 80 β 3 = 77, 77 β 5 = 72, 72 β 7 = 65, 65 β 9 = 56, 56 β 11 = 45, 45 β 13 = 32, 32 β 15 = 17, 17 β 17 = 0
We have subtracted 9 times to get 0.
β81 = 9
(iii) 225 β 1 = 224, 224 β 3 = 221, 221 β 5 = 216, 216 β 7 = 209, 209 β 9 = 200, 200 β 11 = 189, 189 β 13 = 176, 176 β 15 = 161, 161 β 17 = 144, 144 β 19 = 125,
125 β 21 = 104, 104 β 23 = 81, 81 β 25 = 56, 56 β 27 = 29, 29 β 29 = 0
We have subtracted 15 times to get 0.
β225 = 15
Question 14.
Find the square rootofthe following using prime factorisation
(i) 441
(ii) 2025
(iii) 7056
(iv) 4096
Solution:
(i) 441 =Β 3 Γ 3Β ΓΒ 7 Γ 7
β441 = 3 Γ 7 = 21
(ii) 2025 =Β 3 Γ 3Β ΓΒ 3 Γ 3Β ΓΒ 5 Γ 5
β2025 = 3 Γ 3 Γ 5 = 45
(iii) 7056 =Β 2 Γ 2Β ΓΒ 2 Γ 2Β ΓΒ 3 Γ 3Β ΓΒ 7 Γ 7
β7056 = 2 Γ 2 Γ 3 Γ 7 = 84
(iv) 4096 =Β 2 Γ 2Β ΓΒ 2 Γ 2Β ΓΒ 2 Γ 2Β ΓΒ 2 Γ 2Β ΓΒ 2 Γ 2Β ΓΒ 2 Γ 2
β4096 = 2 Γ 2 Γ 2 Γ 2 Γ 2 Γ 2 = 64
Question 15.
Find the least square number which is divisible by each of the number 4, 8 and 12.
Solution:
LCM of 4, 8, 12 is the least number divisible by each of them.
LCM of 4, 8 and 12 = 24
24 =Β 2 Γ 2Β Γ 2 Γ 3
To make it perfect square multiply 24 by the product of unpaired numbers, i.e., 2 Γ 3 = 6
Required number = 24 Γ 6 = 144
Question 16.
Find the square roots of the following decimal numbers
(i) 1056.25
(ii) 10020.01
Solution:
Question 17.
What is the least number that must be subtracted from 3793 so as to get a perfect square? Also, find the square root of the number so obtained.
Solution:
First, we find the square root of 3793 by division method.
Here, we get a remainder 72
612Β < 3793
Required perfect square number = 3793 β 72 = 3721 and β3721 = 61
Question 18.
Fill in the blanks:
(Π°) The perfect square number between 60 and 70 is β¦β¦β¦β¦
(b) The square root of 361 ends with digit β¦β¦β¦β¦..
(c) The sum of first n odd numbers is β¦β¦β¦β¦
(d) The number of digits in the square root of 4096 is β¦β¦β¦..
(e) If (-3)2Β = 9, then the square root of 9 is β¦β¦β¦.
(f) Number of digits in the square root of 1002001 is β¦β¦β¦β¦
(g) Square root ofΒ 36625Β is β¦β¦β¦..
(h) The value of β(63 Γ 28) = β¦β¦β¦β¦
Solution:
(a) 64
(b) 9
(c) n2
(d) 2
(e) Β±3
(f) 4
(g)Β 625
(h) 42
Question 19.
Simplify: β900 + β0.09 + β0.000009
Solution:
We know that β(ab) = βa Γ βb
β900 = β(9 Γ 100) = β9 Γ β100 = 3 Γ 10 = 30
β0.09 = β(0.3 Γ 0.3) = 0.3
β0.000009 = β(0.003 Γ 0.003) = 0.003
β900 + β0.09 + β0.000009 = 30 + 0.3 + 0.003 = 30.303
Squares and Square Roots Class 8 Extra Questions Higher Order Thinking Skills (HOTS)
Question 20.
Find the value of x if
1369βββββ+0.0615+xββββββββββ=37.25
Solution:
Question 21.
Simplify:
Solution:
Question 22.
A ladder 10 m long rests against a vertical wall. If the foot of the ladder is 6 m away from the wall and the ladder just reaches the top of the wall, how high is the wall? (NCERT Exemplar)
Solution:
Let AC be the ladder.
Therefore, AC = 10 m
Let BC be the distance between the foot of the ladder and the wall.
Therefore, BC = 6 m
βABC forms a right-angled triangle, right angled at B.
By Pythagoras theorem,
AC2Β = AB2Β + BC2
102Β = AB2Β + 62
or AB2Β = 102Β β 62Β = 100 β 36 = 64
or AB = β64 = 8m
Hence, the wall is 8 m high.
Question 23.
Find the length of a diagonal of a rectangle with dimensions 20 m by 15 m. (NCERT Exemplar)
Solution:
Using Pythagoras theorem, we have Length of diagonal of the rectangle =Β l2+b2ββββββΒ units
Hence, the length of the diagonal is 25 m.
Question 24.
The area of a rectangular field whose length is twice its breadth is 2450 m2. Find the perimeter of the field.
Solution:
Let the breadth of the field be x metres. The length of the field 2x metres.
Therefore, area of the rectangular field = length Γ breadth = (2x)(x) = (2x2) m2
Given that area is 2450 m2.
Therefore, 2x2Β = 2450
β x2Β = 1225
β x = β1225 or x = 35 m
Hence, breadth = 35 m
and length = 35 Γ 2 = 70 m
Perimeter of the field = 2 (l + b ) = 2(70 + 35) m = 2 Γ 105 m = 210 m.