CHAPTER 5 – EXERCISE 5.3 SOLUTIONS

(i) \(9801\) → \(\sqrt{9801}=99\). One's digit = 9.

(ii) \(99856\) → \(\sqrt{99856}=316\). One's digit = 6.

(iii) \(998001\) → \(\sqrt{998001}=999\). One's digit = 9.

(iv) \(657666025\) → \(\sqrt{657666025}=25645\). One's digit = 5.

Rule used: A perfect square cannot end in digits 2,3,7,8. Also other quick checks (odd/even zeros) apply.

(i) \(153\) → ends with 3 → Not a perfect square.

(ii) \(257\) → ends with 7 → Not a perfect square.

(iii) \(408\) → ends with 8 → Not a perfect square.

(iv) \(441\) → ends with 1 but \(441 = 21^2\) → Perfect square.

Method: Repeatedly subtract successive odd numbers until zero; the number of subtractions = square root.

100:
100 −1 =99 (1), −3 =96 (2), −5 =91 (3), −7 =84 (4), −9 =75 (5), −11 =64 (6), −13 =51 (7), −15 =36 (8), −17 =19 (9), −19 =0 (10).
So \(\sqrt{100}=10\).

169:
169 −1 =168 (1), −3 =165 (2), −5 =160 (3), −7 =153 (4), −9 =144 (5), −11 =133 (6), −13 =120 (7), −15 =105 (8), −17 =88 (9), −19 =69 (10), −21 =48 (11), −23 =25 (12), −25 =0 (13).
So \(\sqrt{169}=13\).

(i) 729
We use prime factorization to find square root.

3	729
3	243
3	81
3	27
3	9
3	3
	1

Thus, \(729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3\).

Grouping: \((3 \times 3)(3 \times 3)(3 \times 3)\).

\(\sqrt{729} = 3 \times 3 \times 3 = 27\).

(ii) 400

2	400
2	200
2	100
2	50
5	25
5	5
	1

\(400 = 2 \times 2 \times 2 \times 2 \times 5 \times 5\).

Grouping: \((2 \times 2)(2 \times 2)(5 \times 5)\).

\(\sqrt{400} = 2 \times 2 \times 5 = 20\).

(iii) 1764

2	1764
2	882
3	441
3	147
7	49
7	7
	1

\(1764 = 2 \times 2 \times 3 \times 3 \times 7 \times 7\).

Grouping: \((2 \times 2)(3 \times 3)(7 \times 7)\).

\(\sqrt{1764} = 2 \times 3 \times 7 = 42\).

(iv) 4096

2	4096
2	2048
2	1024
2	512
2	256
2	128
2	64
2	32
2	16
2	8
2	4
2	2
	1

\(4096 = 2^{12}\).

Grouping: six pairs of 2’s.

\(\sqrt{4096} = 2^6 = 64\).

(v) 7744

2	7744
2	3872
2	1936
2	968
2	484
2	242
2	121
11	121
11	11
	1

\(7744 = 2^6 \times 11^2\).

Grouping: \((2^6)(11^2)\).

\(\sqrt{7744} = 2^3 \times 11 = 88\).

(vi) 9604

2	9604
2	4802
2	2401
7	2401
7	343
7	49
7	7
	1

\(9604 = 2^2 \times 7^4\).

Grouping: \((2 \times 2)(7 \times 7)(7 \times 7)\).

\(\sqrt{9604} = 2 \times 7 \times 7 = 98\).

(vii) 5929

13	5929
13	457
457	—

\(5929 = 77^2\).

Grouping: \(77 \times 77\)

\(\sqrt{5929} = 77\).

(viii) 9216

2	9216
2	4608
2	2304
2	1152
2	576
2	288
2	144
2	72
2	36
2	18
2	9
3	9
3	3
	1

\(9216 = 2^{10} \times 3^2\).

\(\sqrt{9216} = 2^5 \times 3 = 32 \times 3 = 96\).

(ix) 529

23	529
23	23
	1

\(529 = 23 \times 23\).

\(\sqrt{529} = 23\).

(x) 8100

2	8100
2	4050
3	2025
3	675
3	225
3	75
5	25
5	5
	1

\(8100 = 2^2 \times 3^4 \times 5^2\).

\(\sqrt{8100} = 2 \times 3^2 \times 5 = 90\).

(i) 252

2	252
2	126
3	63
3	21
7	7
	1

\(252 = 2 \times 2 \times 3 \times 3 \times 7\).

Here, 7 is unpaired. To make it a perfect square, multiply 252 by 7.

\(252 \times 7 = 1764\), \(\sqrt{1764} = 42\).

(ii) 180

2	180
2	90
3	45
3	15
5	5
	1

\(180 = 2 \times 2 \times 3 \times 3 \times 5\).

Here, 5 is unpaired. To make it a perfect square, multiply 180 by 5.

\(180 \times 5 = 900\), \(\sqrt{900} = 30\).

(iii) 1008

2	1008
2	504
2	252
2	126
3	63
3	21
7	7
	1

\(1008 = 2^4 \times 3^2 \times 7\).

Here, 7 is unpaired. To make it a perfect square, multiply 1008 by 7.

\(1008 \times 7 = 7056\), \(\sqrt{7056} = 84\).

(iv) 2028

2	2028
2	1014
3	507
13	169
13	13
	1

\(2028 = 2^2 \times 3 \times 13^2 \).

Here, 3 is unpaired. To make it a perfect square, multiply 2028 by 3.

\(2028 \times 3 = 6084\), \(\sqrt{6084} = 78\).

(v) 1458

2	1458
3	729
3	243
3	81
3	27
3	9
3	3
	1

\(1458 = 2 \times 3^6\).

Here, 2 is unpaired. To make it a perfect square, multiply 1458 by 2.

\(1458 \times 2 = 2916\), \(\sqrt{2916} = 54\).

(vi) 768

2	768
2	384
2	192
2	96
2	48
2	24
2	12
2	6
3	3
	1

\(768 = 2^8 \times 3\).

Unpaired factor = 3 → Multiply by 3.

\(768 \times 3 = 2304\), \(\sqrt{2304} = 48\).

(i) 252

2	252
2	126
3	63
3	21
7	7
	1

\(252 = 2 \times 2 \times 3 \times 3 \times 7\).

Unpaired factor = 7 → Divide by 252 by 7.

\(252 \div 7 = 36\), Therefore \(\sqrt{36} = 6\).

(ii) 2925

3	2925
3	975
5	325
5	65
13	13
	1

\(2925 = 3^2 \times 5^2 \times 13\).

Unpaired factor = 13 → Divide 2925 by 13.

\(2925 \div 13 = 225\), Therefore \(\sqrt{225} = 15\).

(iii) 396

2	396
2	198
3	99
3	33
11	11
	1

\(396 = 2^2 \times 3^2 \times 11\).

Unpaired factor = 11 → Divide 396 by 11.

\(396 \div 11 = 36\), Therefore \(\sqrt{36} = 6\).

(iv) 2645

5	2645
23	529
23	23
	1

\(2645 = 5 \times 23 \times 23\).

Unpaired factor = 5 → Divide 2645 by 5.

\(2645 \div 5 = 529\), Therefore \(\sqrt{529} = 23\).

(v) 2800

2	2800
2	1400
2	700
2	350
5	175
5	35
7	7
	1

\(2800 = 2^4 \times 5^2 \times 7\).

Unpaired factor = 7 → Divide 2800 by 7.

\(2800 \div 7 = 400\), Therefore \(\sqrt{400} = 20\).

(vi) 1620

2	1620
2	810
3	405
3	135
3	45
3	15
5	5
	1

\(1620 = 2^2 \times 3^4 \times 5\).

Unpaired factor = 5 → Divide 1620 by 5.

\(1620 \div 5 = 324\), Therefore \(\sqrt{324} = 18\).