## ML Aggarwal Class 8 Solutions for ICSE Maths Chapter 3 Squares and Square Roots Check Your Progress

Question 1.

Show that 1089 is a perfect square. Also find the number whose square is 1089.

Solution:

1089 = 3 × 3 × 11 × 11

∵ Prime factors are in pairs and no factor is left.

∴ It is a perfect square and its square root = 3 × 11 = 33

Question 2.

Find the smallest number which should be multiplied by 3675 to make it a perfect square. Also find the square root of this perfect square.

Solution:

3675

Factorising, we get

3675 = 3 × 5 × 5 × 7 × 7

Pairing the same kinds of factors, one factor 3 is left unpaired.

3675 should be multiplied by 3

We get 11025 is a perfect square of 3 × 5 × 7 = 105

Question 3.

Express 121 as the sum of 11 odd numbers.

Solution:

121 = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21

Question 4.

How many numbers lie between 99^{2} and 100^{2}?

Solution:

Total numbers which lie between 99^{2} and 100^{2}

= (99 + 100 – 1) = 198

Question 5.

Write a Pythagorean triplet whose one number is 17.

Solution:

One number of Pythagorean triplet = 17

Let n^{2} + 1 = 17 ⇒ n^{2} = 17 – 1 = 16 = (4)^{2}

∴ n = 4

∴ Numbers will be 2n = 2 × 4 = 8

n^{2} – 1 = 4^{2} – 1 = 15 and n^{2} + 1 = 17

∴ Triplet is (8, 15, 17)

Question 6.

Find the smallest square number which is divisible by each of the numbers 6, 8, 9.

Solution:

Smallest square which is divisible by 6, 8,9

LCM of 6, 8, 9 = 2 × 3 × 3 × 4

= 2 × 2 × 2 × 3 × 3 = 72

Smallest square = 2 × 72 = 144

Question 7.

In an auditorium the number of rows is equal to number of chairs in each row. If the capacity of the auditorium is 1764. Find the number of chairs in each row.

Solution:

In an auditorium, there are

Number of rows = Number of chairs in each row

But, capacity is = 1764 persons

∴ Number of chairs = \(\sqrt{1764}\) = 42

Question 8.

Find the length of diagonal of a rectangle whose length and breadth are 12 m and 5 m respectively.

Solution:

Length of a rectangle = 12 m

and breadth = 5 m

Question 9.

Find the square root of 144 by successive subtraction.

Solution:

Square root of 144 = \(\sqrt{144}\)

144 = 144 – 1 = 143

143 = 143 – 3 = 140

140 = 140 – 5 = 135

135 = 135 – 7 = 128

128 = 128 – 9= 119

119 = 119 – 11 = 108

108 = 108 – 13 = 95

95 = 95 – 15 = 80

80 = 80 – 17 = 63

63 = 63 – 19 = 44

44 – 21 = 23

23 – 23 = 0

∴ Square root = 12

Question 10.

Find the square root of following numbers by prime factorisation:

(i) 5625

(ii) 1521

Solution:

(i) 5625 = 3 × 3 × 5 × 5 × 5 × 5

Square root of 5625 = \(\sqrt{5625}\)

= 3 × 5 × 5 = 75

(ii) 1521 = 3 × 3 × 13 × 13

Square root of 1521 = \(\sqrt{1521}\)

= 3 × 13 = 39

Question 11.

Find the square root of following numbers by long division method:

(i) 21904

(ii) 108241

Solution:

(i) \(\sqrt{21904}\) = 148

(ii) \(\sqrt{108241}\) = 329

Question 12.

Find the square root of following decimal numbers:

(i) 17.64

(ii) 13.3225

Solution:

(i) \(\sqrt{17.64}\) = 4.2

(ii) \(\sqrt{13.3225}\) = 3.65

Question 13.

Find the square root of following fractions:

Solution:

Question 14.

Find the least number which must be subtracted from 2311 to make it a perfect square.

Solution:

2311

Taking square root, we see that 7 is left as remainder.

So, 7 is to be subtracted from 2311.

Question 15.

Find the least number which must be added to 520 to make it a perfect square.

Solution:

520

Taking square root of 520, we see that

(22)^{2} < 520

Then we should take (23)^{2} which is 529

So, 529 – 520 = 9 is to be added

So, that we resultant number will be a perfect

square.

∴ Required number = 9

Question 16.

Find the greatest number of 5 digits which is a perfect square.

Solution:

Greatest 5 digits number = 99999

Taking square root we see that 143 is left as remainder.

So, by subtracting 143 from 99999,

we get the greatest 5 digits which is a perfect square.

Required number = 99999 – 143 = 99856