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Square Roots

Exercise 6.4

Question 1: Find the square root of each of the following numbers by Division method.

(i) 2304

(ii) 4489

(iii) 529

(iv) 3249

(v) 1369

(vi) 5776

(vii) 7921

(viii) 576

(ix) 1024

(x) 3136

(xi) 900

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Question 2: 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

Answer: If there are even number of digits in square then number of digits in

Square Root `=(π)/(2)`

If there are odd number of digits in square then number of digits in

Square Root `=(π+1)/(2)`

(i) 1, (ii) 2, (iii) 2, (iv) 3, (v) 3

3. Find the square root of the following decimal numbers.

(i) 2.56

(ii) 7.29

(iii) 51.84

(iv) 42.25

(v) 31.36

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

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

(ii) 1989

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

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

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

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.

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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

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

Here, `161 xx 1 = 161` is 11 more than 150
So, `1750 + 11 = 1761` is a perfect square

(iii) 252

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

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

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.

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.

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.

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