**Squares and Square Roots**

**Question 1.****Find the square root of each of the following numbers by Division method :(i)** 2304

**(ii)**4489

**(iii)**3481

**(iv)**529

**(v)**3249

**(vi)**1369

**(vii)**5776

**(viii)**7921

**(ix)**576

**(x)**1024

**(xi)**3136

**(xii)**900

**Solution:**

**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

**Solution:**

**Question 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

**Solution:****(i)** Here, the number of decimal places is already even. So, mark off periods and proceed as under :

∴

**(ii)** Here, the number of decimal places are already even. So, mark off periods and proceed as under :

∴

**(iii)** Here, the number of decimal places are already even. So, mark off periods and proceed as under :

∴

**(iv)** Here, the number of decimal places are already even. So, mark off periods and proceed as under :

∴

**(v)** Here, the number of decimal places are already even. So, mark off periods and proceed as under :

∴

**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

**(ii)**1989

**(iii)**3250

**(iv)**825

**(v)**4000

**Solution:****(i)** Let us try to find the square root of 402.

This shows the (20)^{2} is less than 402 by 2. So, in order to get a perfect square, 2 must be subtracted from the given number.

∴ Required perfect square number = 402 – 2 = 400

Also,

**(ii)** Let us try to find the square root of 1989.

This shows that (44)^{2} is less than 1989 by 53. So, in order to get a perfect square, 53 must be subtracted from the given number.

∴ Required perfect square number = 1989 – 53 = 1936

Also,

**(iii)** Let us try to find the square root of 3250.

This shows that (57)^{2} is less than 3250 by 1. So, in order to get a perfect square, 1 must be subtracted from the given number.

∴ Required perfect number = 3250 -1 = 3249

Also,

**(iv)** Let us try to find the square root of 825.

This shows that (28)^{2} is less than 825 by 41. So, in order to get a perfect square, 41 must be subtracted from the given number.

∴ Required perfect square number = 825 – 41 = 784

Also,

**(v)** Let us try to find the square root of 4000.

This shows that (63)^{2} is less than 4000 by 31. So, in order to get a perfect square, 31 must be subtracted from the given number.

∴ Required perfect square number = 4000 – 31 = 3969

Also,

**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

**(ii)**1750

**(iii)**252

**(iv)**1825

**(v)**6412

**Solution:****(i)** We try to find out the square root of 525.

**(ii)** We try to find out the square root of 1750.

**(iii)** We try to find out the square root of 252.

**(iv)** We try to find out the square root of 1825.

**(v)** We try to find out the square root of 6412.

**Question 6.****Find the length of the side of a square whose area is 441 m ^{2}.**

**Solution:**

**Question 7.****In a right triangle ABC, ∠B = 90°.(a)** If AB = 6 cm, BC = 8 cm, find AC.

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

**Solution:**

**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.**

**Solution:**

Let us find the square root of 1000.

This shows that (31)^{2} is less than 1000 by 39 and (32)^{2} =1024. Thus, the gardener needs 1024 -1000 = 24 plants more to plant in such a way that the number of rows and the number of columns remain the same.

**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?**

**Solution:**

Let us find the square root of 500.

This shows that (22)^{2} = 484 is less than 500 by 16.

∴ 16 students have to go out for others to do the P.T. practice as per condition.