Home > Pre-Algebra calculators > Square root of a number using long division method example

7. Square root by long division method example ( Enter your problem )
  1. Example-1
Other related methods
  1. Square of a number
  2. Cube of a number
  3. Nth Power of a number
  4. Square root by prime factorization method
  5. Cube root by prime factorization method
  6. Nth root by prime factorization method
  7. Square root by long division method
  8. Cube root by long division method
  9. Babylonian method for Square root
  10. Babylonian method for Cube root
  11. Find the smallest number which must be Added / Substracted / Multiplied / Divided to 180 to make it perfect Square / Cube
  12. Find Least number of 4 digits which is a perfect Square

6. Nth root by prime factorization method
(Previous method)
8. Cube root by long division method
(Next method)

1. Example-1





1. Find Square root of 4096 using long division method

Solution:
 64  
64096  
636   
124 496  
4 496  
128 0  

 
Dividend = 4096
Square Root = 64
Remainder = 0
2. Find Square root of 70 using long division method

Solution:
Solution
 8.3666  
870.00000000  
64      
163 600     
489     
1666 11100    
9996    
16726 110400   
100356   
167326 1004400  
1003956  
167332 444  

 
Number = 70.00000000
Square Root = 8.3666


Step by step solution :

Step-1 :
Make pair of digits of given number starting with digit at one's place. Put bar on each pair.
    
70  

Step-2 :
Now we have to multiply a number by itself such that the product `<=` 70

Here `8 xx 8=64<= 70`, So divisor is 8 and quotient is 8. Now do the division and get the remainder.

 8  
870  
64  
6  

Step-3 :
Now , we have to bring down 00 and quotient 8 is multiplied by 2 becomes 16, which is starting digit of new divisor
 8.   
870.00  
64   
16 600  

Step-4 :
3 should be the digit at one's place of new divisor because when 163 is multiplied by 3 we get 489.
So new divisor is 163 and next digit of quotient is 3. Now do the division and get the remainder.
 8.3  
870.00  
64   
163 600  
489  
111  

Step-5 :
Now , we have to bring down 00 and quotient 83 is multiplied by 2 becomes 166, which is starting digit of new divisor
 8.3   
870.0000  
64    
163 600   
489   
166 11100  

Step-6 :
6 should be the digit at one's place of new divisor because when 1666 is multiplied by 6 we get 9996.
So new divisor is 1666 and next digit of quotient is 6. Now do the division and get the remainder.
 8.36  
870.0000  
64    
163 600   
489   
1666 11100  
9996  
1104  

Step-7 :
Now , we have to bring down 00 and quotient 836 is multiplied by 2 becomes 1672, which is starting digit of new divisor
 8.36   
870.000000  
64     
163 600    
489    
1666 11100   
9996   
1672 110400  

Step-8 :
6 should be the digit at one's place of new divisor because when 16726 is multiplied by 6 we get 100356.
So new divisor is 16726 and next digit of quotient is 6. Now do the division and get the remainder.
 8.366  
870.000000  
64     
163 600    
489    
1666 11100   
9996   
16726 110400  
100356  
10044  

Step-9 :
Now , we have to bring down 00 and quotient 8366 is multiplied by 2 becomes 16732, which is starting digit of new divisor
 8.366   
870.00000000  
64      
163 600     
489     
1666 11100    
9996    
16726 110400   
100356   
16732 1004400  

Step-10 :
6 should be the digit at one's place of new divisor because when 167326 is multiplied by 6 we get 1003956.
So new divisor is 167326 and next digit of quotient is 6. Now do the division and get the remainder.
 8.3666  
870.00000000  
64      
163 600     
489     
1666 11100    
9996    
16726 110400   
100356   
167326 1004400  
1003956  
444  

3. Find Square root of 28900 using long division method

Solution:
Solution
 170  
1 28900  
1    
27 189   
189   
340 0  
0  
340 0  

 
Number = 28900
Square Root = 170


Step by step solution :

Step-1 :
Make pair of digits of given number starting with digit at one's place. Put bar on each pair.
      
28900  

Step-2 :
Now we have to multiply a number by itself such that the product `<=` 2

Here `1 xx 1=1<= 2`, So divisor is 1 and quotient is 1. Now do the division and get the remainder.

 1    
1 28900  
1    
1    

Step-3 :
Now , we have to bring down 89 and quotient 1 is multiplied by 2 becomes 2, which is starting digit of new divisor
 1    
1 28900  
1    
2 189   

Step-4 :
7 should be the digit at one's place of new divisor because when 27 is multiplied by 7 we get 189.
So new divisor is 27 and next digit of quotient is 7. Now do the division and get the remainder.
 17   
1 28900  
1    
27 189   
189   
0   

Step-5 :
Now , we have to bring down 00 and quotient 17 is multiplied by 2 becomes 34, which is starting digit of new divisor
 17   
1 28900  
1    
27 189   
189   
34 0  

Step-6 :
0 should be the digit at one's place of new divisor because when 340 is multiplied by 0 we get 0.
So new divisor is 340 and next digit of quotient is 0. Now do the division and get the remainder.
 170  
1 28900  
1    
27 189   
189   
340 0  
0  
0  

4. Find Square root of 101 using long division method

Solution:
Solution
 10.0498  
1 101.00000000  
1       
20 1      
0      
200 100     
0     
2004 10000    
8016    
20089 198400   
180801   
200988 1759900  
1607904  
200996 151996  

 
Number = 101.00000000
Square Root = 10.0498


Step by step solution :

Step-1 :
Make pair of digits of given number starting with digit at one's place. Put bar on each pair.
     
101  

Step-2 :
Now we have to multiply a number by itself such that the product `<=` 1

Here `1 xx 1=1<= 1`, So divisor is 1 and quotient is 1. Now do the division and get the remainder.

 1   
1 101  
1   
0   

Step-3 :
Now , we have to bring down 01 and quotient 1 is multiplied by 2 becomes 2, which is starting digit of new divisor
 1   
1 101  
1   
2 1  

Step-4 :
0 should be the digit at one's place of new divisor because when 20 is multiplied by 0 we get 0.
So new divisor is 20 and next digit of quotient is 0. Now do the division and get the remainder.
 10  
1 101  
1   
20 1  
0  
1  

Step-5 :
Now , we have to bring down 00 and quotient 10 is multiplied by 2 becomes 20, which is starting digit of new divisor
 10.   
1 101.00  
1    
20 1   
0   
20 100  

Step-6 :
0 should be the digit at one's place of new divisor because when 200 is multiplied by 0 we get 0.
So new divisor is 200 and next digit of quotient is 0. Now do the division and get the remainder.
 10.0  
1 101.00  
1    
20 1   
0   
200 100  
0  
100  

Step-7 :
Now , we have to bring down 00 and quotient 100 is multiplied by 2 becomes 200, which is starting digit of new divisor
 10.0   
1 101.0000  
1     
20 1    
0    
200 100   
0   
200 10000  

Step-8 :
4 should be the digit at one's place of new divisor because when 2004 is multiplied by 4 we get 8016.
So new divisor is 2004 and next digit of quotient is 4. Now do the division and get the remainder.
 10.04  
1 101.0000  
1     
20 1    
0    
200 100   
0   
2004 10000  
8016  
1984  

Step-9 :
Now , we have to bring down 00 and quotient 1004 is multiplied by 2 becomes 2008, which is starting digit of new divisor
 10.04   
1 101.000000  
1      
20 1     
0     
200 100    
0    
2004 10000   
8016   
2008 198400  

Step-10 :
9 should be the digit at one's place of new divisor because when 20089 is multiplied by 9 we get 180801.
So new divisor is 20089 and next digit of quotient is 9. Now do the division and get the remainder.
 10.049  
1 101.000000  
1      
20 1     
0     
200 100    
0    
2004 10000   
8016   
20089 198400  
180801  
17599  

Step-11 :
Now , we have to bring down 00 and quotient 10049 is multiplied by 2 becomes 20098, which is starting digit of new divisor
 10.049   
1 101.00000000  
1       
20 1      
0      
200 100     
0     
2004 10000    
8016    
20089 198400   
180801   
20098 1759900  

Step-12 :
8 should be the digit at one's place of new divisor because when 200988 is multiplied by 8 we get 1607904.
So new divisor is 200988 and next digit of quotient is 8. Now do the division and get the remainder.
 10.0498  
1 101.00000000  
1       
20 1      
0      
200 100     
0     
2004 10000    
8016    
20089 198400   
180801   
200988 1759900  
1607904  
151996  





This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then Submit Here



6. Nth root by prime factorization method
(Previous method)
8. Cube root by long division method
(Next method)





Share this solution or page with your friends.
 
 
Copyright © 2025. All rights reserved. Terms, Privacy
 
 

.