1. Find the value of √5 

To find square root of 5 up to desire number of decimal places p, we put 2(p + 1) number of zero after the decimal place.
Then we find the square root of this number up to p + 1 places of decimal.
Then we round up the values up to pplaces of decimal.

2. Find the value of √45369.

       First we have to separate the digits in to two each from right
       ie) 4, 53 , 69
       Then find a square no. less than or equal to 4
       4 is the square number .
      √4 = 2 
     Write 4 below 4  and subtract. Write 2 at the top.
    Then bring 53 to down, and write double of 2 ie) 4 below 2 and left of 53
    Then find a number x such that (4x)  X (x) , so that it is less than 53.
    Then write 41 below  53 and subtract we get 12
    Write 1 in right of 2 in the top. Then bring down 69 below and write along with 12.
    Then write double of 21 ie) 42 below 41 and left of 1269.
    Choose a number x such that 42x X x which is less than or equals to 1269
    Then we get 3 is that number. write 3 in the right of 21. Therefore 213 is the square root of 45369.

     
                 √45369 = 213.  

3. Find the value of √434281.

      First we have to separate the digits in to two each from right, ie) 43, 42 ,81

     Then find a square no. less than or equal to 43

     36 is the square number .

    √36 = 6

    Write 36 below 43  and subtract. write 2 at the top.

     Then bring 42 to down, write along with 7

     And write double of 6 ie) 12 below 6 and left of 742. Then find a number x such that (12x)  X (x) , so that it is less than 742.

     Then write 125 below  6 and write 625 below 742 subtract we get 117

     Write 5 at the top near 6. Then bring down 81 and write along with 117

     Then write double of 65 ie) 130 below 125 and left of 11781

      Choose a number x such that 130x X x which is less than or equals to 11781.

      Then we get 9 is that number. write 9 in the right of 65.

      Therefore 659 is the square root of 434281.

     
       √434281= 659.

4. Find the square root of 1156 by prime factorization.
      Resolving 1156 into prime factors, we get
      1156 = 2 x 2 x 17 x 17
              = 22  x 172
      √1156 = ( 2 x 17 )
                    = 34.
      

5. Find the square root of 2401.
     2401 = 7 x 7 x 7 x 7
    √ 2401 = 7²
               = 49 .
    

6. Find the square root by factorization method 18496.
     18496 = 2 x 2 x 2 x2 x 2 x 2 x 17 x 17
               = 2 x 2 x 2 x 17
               = 8 x 17
               = 136. 
         

7. Find the square of 289.
      ( 289 ) ² = 83521.
      

8. Find the square of 321.
       321 ² = 103041 
      

9. Find the square root of 226576.
       √ 226576  = 476.
       

10. Find the square of 936.
          936² = 936 x 936 = 876096.
        

11. If Area of square is 576 cm².  Then one side of the square is ______ unit ? 
        Given  that area a² = 567 cm²
        One side of the square a, = √567  cm
        576 = 2 x 2 x 2 x 2 x 2 x2 x 3 x3
        √576 = 2 x 2x 2x3
                    = 8 x 3 = 24 cm
       

12. In an assembly there area 1024 students.  If number of columns and number of rows are equal, then the number of student in each row is ?
        Total number of students = 1024.
        Let a be the number of student in each row.
        Since, number of columns and number of rows are equal, a² = 1024.
        a = √1024
      1024 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2x  2 x 2 
       Therefore √1024 = 2 x 2 x 2x 2x 2
                                = 1024.
      

13. Find the smallest number by which 252 must be multiplied so that the product becomes a perfect square. Also find the square root of the perfect square so obtained.

     Writing 252 as its prime factors we get 252 = 2 × 2 × 3 × 3 × 7. The prime factors 2 and 3 occur in pairs but prime factor 7 occurs alone.     Therefore 252 must be multiplied with 7 to get a perfect square number

    ∴ New number =252 × 7=1764

     Now 1764=(2 × 2) × (3 × 3) × (7 × 7).

      √1764 = 2 × 3 × 7 = 42.

14. Find the smallest square number which is divisible by each of the number 4, 6 and 12.

       We know that smallest number which is divisible by each of the number 4, 6, 12, is L C M (4,6,12) Now L C M (4,6,12) is 12.

       By resolving 12 into its prime factors we find 12 = (2 × 2) × 3, Since 3 occurs alone.

       ∴ To make 12 a perfect square we have to multiply it with 3.

       ∴ Small square number divisible by 4,6,12 is 36.

15. Find the least number which must be subtracted from 18315 to make it a perfect square. Also find the square root of the resulting number.

        Here we find that remainder in the last step is 90, it means 18315 is 90 more than the square of 135. So we must subtract 90 from 18315 to make it a perfect square.

        ∴ Perfect square number 18315 - 90 = 18225 and √18225 = 135.

16. Find the greatest number of five digits which is a perfect square.

       Greatest five digit number is 99999. As 99999 is not a perfect square, we must first find the smallest number to be subtracted from 99999 to make it a perfect square. So we apply method of long division on 99999.

       ∴ We must subtract 143 from 99999 to get largest five digit number which is a perfect square.

       ∴ Required number = 99999 - 143 = 99856.

17. Fill in the blanks.

a. The difference of the square of two consecutive natural numbers is equal to their sum.
b. The number of zeroes at the end of a perfect square ending with zeroes, is always even.
c. Square numbers are always positive.
d. There are four digits in the square root of 2540836.
e. In a and b are natural numbers such that a=b², then a is a perfect square.

18. Match the Following

a. Square root is the inverse operation of                            Eight odd numbers
b. Square numbers can only have                                       Pythagorean triplets
c. Squares of even numbers are                                         Even number of zeros at the end
d. 64 can be written as the sum of                                      Even
e. If 42 + 32 = 52, then 3, 4 and 5 are called                       Sqaure 

Ans:

a. Square root is the inverse operation of                    Sqaure
b. Square numbers can only have                               Even number of zeros at the end
c. Squares of even numbers are                                  Even
d. 64 can be written as the sum of                               Eight odd numbers
e. If 42 + 32 = 52, then 3, 4 and 5 are called               Pythagorean triplets