Square Roots

Exercise 6.3 Part 1

Question 1: What could be the possible ‘one’s’ digits of the square root of each of the following numbers?

(i) 9801

Answer: 1 and 9.
Explanation: Since 1² and 9² give 1 at unit’s place, so these are the possible values of unit digit of the square root.

(ii) 99856

Answer: 4 and 6
Explanation: Since, 4²= 16 and 6²= 36, hence, 4 and 6 are possible digits

(iii) 998001

Answer: 1 and 9

(iv) 657666025

Answer: 5
Explanation: Since, 5²= 25, hence 5 is possible.

Question 2: Without doing any calculation, find the numbers which are surely not perfect squares.

(i) 153 (ii) 257 (iii) 408 (iv) 441

Answer: (i) 153 (ii) 257 (iii) 408
Explanation: Since, (i), (ii) and (iii) are surely not be perfect square as these numbers end with 3, 7 and 8. A number can be a perfect square if it ends with 0, 1, 4, 5, 6, 9 only

Question 3: Find the square roots of 100 and 169 by the method of repeated subtraction.

Answer: Square root of 100 by Repeated subtraction:

1. 100 - 1 = 99
2. 99 - 3 = 96
3. 96 - 5 = 91
4. 91 -7 = 84
5. 84 - 9 = 75
6. 75 - 11 = 64
7. 64 - 13 = 51
8. 51 - 15 = 3610. 19 - 19 = 0
We get 0 at 10th step. Thus, 10 is the square root of 100.

So, `sqrt(100)=100`

Square root of 169 by Repeated subtraction:

1. 169 - 1 = 168
2. 168 - 3 = 165
3. 165 - 5 = 160
4. 160 - 7 = 153
5. 153 - 9 = 144
6. 144 - 11 = 133
7. 133 - 13 = 120
8. 120 - 15 = 105
9. 105 - 17 = 88
10. 88 - 19 = 69
11. 69 - 21 = 48
12. 48 - 23 = 25
13. 25 - 25 = 0
We get 0 at 13th step. Thus 13 is the square root of 169

So, `sqrt(169)=13`

Question 4: Find the square roots of the following numbers by the Prime Factorisation Method.

(i) 729

3	729
3	243
3	81
3	27
3	9
	3

So, `729=3^6`

Or, `sqrt(729)=3^3=27`

(ii) 400

2	400
2	200
2	100
2	50
5	25
	5

So, `400=2^4×5^2`

Or, `sqrt(400)=2^2×5=20`

(iii) 1764

Answer:

2	1764
2	882
3	441
3	147
7	49
	7

`1764=2^2xx3^2xx7^2`

`sqrt(1764)=2xx3xx7=42`

(iv) 4096

Answer:

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

`4096=2^(12)`

`sqrt(4096)=2^6=64`

(v) 7744

Answer:

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

`7744=2^6xx11^2`

`sqrt(7744)=2^3xx11=88`

(vi) 9604

Answer:

2	9604
2	4802
7	2401
7	343
7	49
	7

`9604=2^2xx7^4`

`sqrt(9604)=2xx7^2=98`

(vii) 5929

Answer:

11	5929
11	539
7	49
	7

`5929=7^2xx11^2`

`sqrt(5929)=7xx11=77`

(viii) 9216

Answer:

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

`9216=2^(10)xx3^2`

`sqrt(9216)=2^5xx3=96`

(ix) 529

Answer: `529=23xx23`

`sqrt(529)=23`

(x) 8100

Answer:

2	8100
2	4050
5	2025
5	405
3	81
3	27
3	9
	3

`8100=2^2xx3^4xx5^2`

`sqrt(8100)=2xx3^2xx5=90`