Square Roots

Exercise 6.4

Part 2

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

2 2	402 4	2
4	002

It is clear that if 2 is subtracted then we will get 400, which is a perfect square.

(ii) 1989

4 4	1989 16	4
8	389

Here, `84 xx 4 = 336` which is less than 389
And, `85 xx 5 = 425`, which is more than 389
Hence the required difference `= 389 - 336 = 53`
`1989 - 53 = 1936` is a perfect square.

(iii) 3250

5 5	3250 25	5
10	750

Here, `107 xx 7 = 749` is less than 750
`108 xx 8 = 864` is more than 750
Hence, the required difference `= 750 - 749 = 1`
`3250 - 1 = 3249` is a perfect square.

(iv) 825

2 2	825 4	2
4	425

Here, `48 xx 8 = 384` is less than 425
`49 xx 9 = 441` is more than 425
Hence, the required difference `= 425 - 384 = 41`
`825 - 41 = 784` is a perfect square.

(v) 4000

6 6	4000 36	6
12	400

Here, `123 xx 3 = 369` is less than 400
`124 xx 4 = 496` is more than 400
Hence, the required difference `= 400 - 369 = 31`
`4000 - 31 = 3969` is a perfect square.

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

2 2	525 4	2
4	125

Here, `43 xx 3 = 129` is more than 125
`42 xx 2 = 84` is less than 125
Hence, required addition `= 129 - 125 = 4`
`525 + 4 = 529` is a perfect square.

(ii) 1750

4 4	1750 16	4
8	150

Here, `81xx1=81` is less 150

And, `82xx2=164` which is 14 more than 150

So, `1750+14=1764` is a perfect square

(iii) 252

1 1	252 1	1
2	152

Here, `25 xx 5 = 125` is less than 152
`26 xx 6 = 156` is more than 152
Required difference `= 156 - 152 = 4`
So, `252 + 4 = 256` is a perfect square

(iv) 1825

4 4	1825 16	4
8	225

Here, `82 xx 2 = 164` is less than 225
`83 xx 3 = 249` is more than 225
Required difference `= 249 - 225 = 24`
So, `1825 + 24 = 1849` is a perfect square

(v) 6412

8 8	6412	8
16	12

Here, we need `161 xx 1 = 161`
Required difference `= 161 - 12 = 149`
So, `6412 + 149 = 6561` is a perfect square

Question 6: Find the length of the side of a square whose area is 441 m².

Answer: Area of Square = Side²

Side `=sqrt\text(Area)`

`441=3xx3xx7xx7`

Or, `sqrt(441)=3xx7=21`

(a) If AB = 6 cm, BC = 8 cm, find AC

Answer: `= AC^2= AB^2 + BC^2`
`= 6^2 + 8^2 = 36 + 64 = 100`

`AC=sqrt(100)=10`

(b) If AC = 13 cm, BC = 5 cm, find AB

Answer: `AB^2= AC^2- BC^2`
`= 13^2- 5^2= 169 - 25 = 144`

`AB=sqrt(144)=12`

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 same. Find the minimum number of plants he needs more for this.

3 3	1000 9	3
6	100

Here, `61xx1=61` is less than 100
`62xx2=124` is more than 100
Hence, the required difference `= 100-61=39`
Min. number of plants required `= 1000-39=961`

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 number of columns. How many children would be left out in this arrangement.

2 2	500 4	2
4	100

Here, `42 xx 2 = 84` is less than 100
`43 xx 3 = 129` is more than 100
Hence, the required difference `= 100 - 84 = 16`
So, 16 children will be left out in the arrangement.