8th Maths 6.4
Chapter 6
Squares and Square Roots
NCERT Class 8th solution of Exercise 6.1
NCERT Class 8th solution of Exercise 6.2
NCERT Class 8th solution of Exercise 6.3
Exercise 6.4
Q1. Find the square root of each of the following numbers by 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
Sol.
i) 2304
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Answer:
√2304=48
ii) 4489
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Answer:
√4489=67
iii) 3481
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Answer:
√3481=59
iv) 529
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Answer:
√529=23
v) 3249
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Answer:
√3249=57
vi) 1369
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Answer:
√1369=37
vii) 5776
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Answer:
√5776=76
viii) 7921
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Answer:
√7921=89
ix) 576
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Answer:
√576=24
x) 1024
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Answer:
√1024=32
xi) 3136
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Answer:
√3136=56
xii) 900
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Answer:
√900=30
Q2. 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
Sol.
We know that, A perfect square is of n-digits, then its square root will have n2 digit if n is even or n+12 if n is odd
i) 64
n=2 [n is even]
=n2
=22
=1
Answer:
Number of digit in √64=1
ii) 144
n=3 [n is odd]
=n+12
=3+12
=42
=2
Answer:
Number of digit in √144=2
iii) 4489
n=4 [n is even]
=n2
=42
=2
Answer:
Number of digit in √4489=2.
iv) 27225
n=5 [n is odd]
=n+12
=5+12
=62
=3
Answer:
Number of digit in √27225=3
v) 390625
n=6 [n is even]
=n2
=62
=3
Answer:
Number of digit in √390625=3.
Q3. Find the square root of the following decimal numbers.
i) 2.56
ii) 7.29
iii) 51.84
iv) 42.25
v) 31.36
Sol.
i) 2.56
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Answer:
√2.56=1.6
ii) 7.29
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Answer:
√7.29=2.7
iii) 51.84
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Answer:
√51.84=7.2
iv) 42.25
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Answer:
√42.25=6.5
v) 31.36
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Answer:
√31.36=5.6
Q4. 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
Sol.
i)
By long division method
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We get remainder 2.
Its shows that 202 is less than 402 by 2.
2 is the least number which must be subtracted from the 402 to get a perfect square.
Therefore perfect square is 402 - 2 = 400.
And√400=20.
Answer:
2;20.
ii)
By long division method
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We get remainder 53.
Its shows that 442 is less than 1989 by 53.
53 is the least number which must be subtracted from the 1989 to get a perfect square.
Therefore perfect square is 1989 - 53 = 1936.
And√1936=44.
Answer:
53;44.
iii)
By long division method
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We get remainder 1.
Its shows that 572 is less than 3250 by 1.
2 is the least number which must be subtracted from the 3250 to get a perfect square.
Therefore perfect square is 3250 - 1 = 3249.
And√3250=57.
Answer:
1;57.
iv)
By long division method
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We get remainder 41.
Its shows that 282 is less than 825 by 41.
41 is the least number which must be subtracted from the 825 to get a perfect square.
Therefore perfect square is 825 - 41 = 784.
And√784=28.
Answer:
41;28.
v)
By long division method
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We get remainder 31.
Its shows that 632 is less than 4000 by 31.
31 is the least number which must be subtracted from the 4000 to get a perfect square.
Therefore perfect square is 4000 - 31 = 3969.
And√3969=63.
Answer:
31;63.
Q5. 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
Sol.
i)
By division method
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We get remainder 41.
It shows that 222is less than 525 by 41.
Next perfect square number is 232=529.
Hence, the number to be added is 529-525=4.
Answer:
4;23.
ii)
By division method
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We get remainder 69.
It shows that 412is less than 1750 by 69.
Next perfect square number is 422=1764.
Hence, the number to be added is 1764-1750=14.
Answer:
14;42.
iii)
By division method
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We get remainder 27.
It shows that 152is less than 252 by 27.
Next perfect square number is 162=256.
Hence, the number to be added is 256-252=4.
Answer:
4;16.
iv)
By division method
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We get remainder 61.
It shows that 422is less than 1825 by 61.
Next perfect square number is 432=1849.
Hence, the number to be added is 1849-1825=24.
Answer:
24;43.
v)
By division method
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We get remainder 12.
It shows that 802is less than 6412 by 12.
Next perfect square number is 812=6561.
Hence, the number to be added is 6561-6412=149.
Answer:
149;81.
Q6. Find the length of the side of a square whose area is 441m2.
Sol.
Let the length of the side of a square is x
Then
Area of square = (Side)2
441=(x)2
441=x×x
441=x2
x=
text{By long division method}
text{x} = 21
text{Answer:}
text{The length of the side of a square is 21 m}^2.
Q7. In a right triangle text{ABC, } angle text{B} = 90^circ.
a) If text{AB = 6 cm, BC = 8 cm,} find text{AC}
b) If text{AC = 13 cm, BC = 5 cm,} find text{AB}
text{Sol.}
a)
text{In right }triangletext{ ABC}
text{By Pythagors Theorem}
text{AC}^2 = text{AB}^2 + text{BC}^2
text{AC}^2 = (6)^2 + (8)^2
text{AC}^2 = 36 + 64
text{AC}^2 = 100
text{AC} = sqrt100
text{By long division method}
text{AC} = 10
text{Answer:}
text{AC} = 10.
b)
text{In right }triangletext{ ABC}
text{By Pythagors Theorem}
text{AC}^2 = text{AB}^2 + text{BC}^2
text{AB}^2 = text{AC}^2 - text{BC}^2
text{AB}^2 = (13)^2 - (5)^2
text{AB}^2 = 169 - 25
text{AB}^2 = 144
text{AB} = sqrt144
text{By long division method}
text{AB} = 12
text{Answer:}
text{AB} = 12.
Q8. 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 same. Find the minimum number of plants he needs more for this.
text{Sol.}
text{Let the number of rows be x}
text{and the number of columns be x}
text{Then x}times text{x} =1000
text{x}^2 = 1000
text{x} = sqrt1000
text{We find }sqrt1000 text{by long division method}
text{The remainder is 39}
text{This shows that the square of 31 is less than} 1000
text{Next perfect square number is 32 is equal}text{ to 1024}
text{Hence, the number to be added is}text{ 1024 - 1000 = 24}
text{Answer:}
text{The minimum number of plants he needs 24}
Q9. 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 number of columns. How many children would be left out in this arrangment.
text{Sol.}
text{Let the number of rows be x}
text{and the number of columns be x}
text{Then x}times text{x } = 500
text{x}^2 = 500
text{x} = sqrt500
text{By long division method}
text{remainder is 16}
500 - 16 = 484
text{Answer:}
text{16 children would be left out in this}text{ arrangment}.
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