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Home > Pre-Algebra calculators > Cube root of a number using long division method example
8. Cube root by long division method example ( Enter your problem )
( 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
7. Square root by long division method
(Previous method)		9. Babylonian method for Square root
(Next method)

1. Example-1


1. Find Cube root of 4096 using long division method

Solution:
 16  
1 4096  
1 1   
516 3096 12× 300 + 1 × 30 × 6 + 62 = 300 + 180 + 36 = 516
6 3096  
--- 0  

 
Dividend = 4096
CubeRoot = 16
Remainder = 0
2. Find Cube root of 2 using long division method

Solution:
Solution
 1.2599  
1 2.000000000000  
1      
364 1000    `300 xx 1^2 + 30 xx 1 xx 2 + 2^2=300 + 60 + 4=364`
728     
45025 272000   `300 xx 12^2 + 30 xx 12 xx 5 + 5^2=43200 + 1800 + 25=45025`
225125    
4721331 46875000  `300 xx 125^2 + 30 xx 125 xx 9 + 9^2=4687500 + 33750 + 81=4721331`
42491979   
475864311 4383021000 `300 xx 1259^2 + 30 xx 1259 xx 9 + 9^2=475524300 + 339930 + 81=475864311`
4282778799  
--- 100242201  

 
Number = 2.000000000000
Cube Root = 1.2599


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.
    
2  

Step-2 :
Now leftmost digits is 2. Now find the largest number whose cube is `<=` 2

It is 1, whose cube is 1. Write 1 as quotient and subtract 1
 1  
1 2  
1  
1  

Step-3 :
Now, we have to bring down the next three digits 000
By trial and error, next quotient digit is 2
`300 xx 1^2 + 30 xx 1 xx 2 + 2^2=300 + 60 + 4=364`

So our new divisor is 364
 1.   
1 2.000  
1   
364 1000 `300 xx 1^2 + 30 xx 1 xx 2 + 2^2=300 + 60 + 4=364`


Step-4 :
Now multiply 364 by 2 (`364 xx 2=728`) and subtract it.

 1.2  
1 2.000  
1   
364 1000 `300 xx 1^2 + 30 xx 1 xx 2 + 2^2=300 + 60 + 4=364`
728  
272  


Step-5 :
Now, we have to bring down the next three digits 000
By trial and error, next quotient digit is 5
`300 xx 12^2 + 30 xx 12 xx 5 + 5^2=43200 + 1800 + 25=45025`

So our new divisor is 45025
 1.2   
1 2.000000  
1    
364 1000  `300 xx 1^2 + 30 xx 1 xx 2 + 2^2=300 + 60 + 4=364`
728   
45025 272000 `300 xx 12^2 + 30 xx 12 xx 5 + 5^2=43200 + 1800 + 25=45025`


Step-6 :
Now multiply 45025 by 5 (`45025 xx 5=225125`) and subtract it.

 1.25  
1 2.000000  
1    
364 1000  `300 xx 1^2 + 30 xx 1 xx 2 + 2^2=300 + 60 + 4=364`
728   
45025 272000 `300 xx 12^2 + 30 xx 12 xx 5 + 5^2=43200 + 1800 + 25=45025`
225125  
46875  


Step-7 :
Now, we have to bring down the next three digits 000
By trial and error, next quotient digit is 9
`300 xx 125^2 + 30 xx 125 xx 9 + 9^2=4687500 + 33750 + 81=4721331`

So our new divisor is 4721331
 1.25   
1 2.000000000  
1     
364 1000   `300 xx 1^2 + 30 xx 1 xx 2 + 2^2=300 + 60 + 4=364`
728    
45025 272000  `300 xx 12^2 + 30 xx 12 xx 5 + 5^2=43200 + 1800 + 25=45025`
225125   
4721331 46875000 `300 xx 125^2 + 30 xx 125 xx 9 + 9^2=4687500 + 33750 + 81=4721331`


Step-8 :
Now multiply 4721331 by 9 (`4721331 xx 9=42491979`) and subtract it.

 1.259  
1 2.000000000  
1     
364 1000   `300 xx 1^2 + 30 xx 1 xx 2 + 2^2=300 + 60 + 4=364`
728    
45025 272000  `300 xx 12^2 + 30 xx 12 xx 5 + 5^2=43200 + 1800 + 25=45025`
225125   
4721331 46875000 `300 xx 125^2 + 30 xx 125 xx 9 + 9^2=4687500 + 33750 + 81=4721331`
42491979  
4383021  


Step-9 :
Now, we have to bring down the next three digits 000
By trial and error, next quotient digit is 9
`300 xx 1259^2 + 30 xx 1259 xx 9 + 9^2=475524300 + 339930 + 81=475864311`

So our new divisor is 475864311
 1.259   
1 2.000000000000  
1      
364 1000    `300 xx 1^2 + 30 xx 1 xx 2 + 2^2=300 + 60 + 4=364`
728     
45025 272000   `300 xx 12^2 + 30 xx 12 xx 5 + 5^2=43200 + 1800 + 25=45025`
225125    
4721331 46875000  `300 xx 125^2 + 30 xx 125 xx 9 + 9^2=4687500 + 33750 + 81=4721331`
42491979   
475864311 4383021000 `300 xx 1259^2 + 30 xx 1259 xx 9 + 9^2=475524300 + 339930 + 81=475864311`


Step-10 :
Now multiply 475864311 by 9 (`475864311 xx 9=4282778799`) and subtract it.

 1.2599  
1 2.000000000000  
1      
364 1000    `300 xx 1^2 + 30 xx 1 xx 2 + 2^2=300 + 60 + 4=364`
728     
45025 272000   `300 xx 12^2 + 30 xx 12 xx 5 + 5^2=43200 + 1800 + 25=45025`
225125    
4721331 46875000  `300 xx 125^2 + 30 xx 125 xx 9 + 9^2=4687500 + 33750 + 81=4721331`
42491979   
475864311 4383021000 `300 xx 1259^2 + 30 xx 1259 xx 9 + 9^2=475524300 + 339930 + 81=475864311`
4282778799  
100242201  
3. Find Cube root of 5 using long division method

Solution:
Solution
 1.7099  
1 5.000000000000  
1      
559 4000    `300 xx 1^2 + 30 xx 1 xx 7 + 7^2=300 + 210 + 49=559`
3913     
86700 87000   `300 xx 17^2 + 30 xx 17 xx 0 + 0^2=86700 + 0 + 0=86700`
0    
8715981 87000000  `300 xx 170^2 + 30 xx 170 xx 9 + 9^2=8670000 + 45900 + 81=8715981`
78443829   
876665811 8556171000 `300 xx 1709^2 + 30 xx 1709 xx 9 + 9^2=876204300 + 461430 + 81=876665811`
7889992299  
--- 666178701  

 
Number = 5.000000000000
Cube Root = 1.7099


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.
    
5  

Step-2 :
Now leftmost digits is 5. Now find the largest number whose cube is `<=` 5

It is 1, whose cube is 1. Write 1 as quotient and subtract 1
 1  
1 5  
1  
4  

Step-3 :
Now, we have to bring down the next three digits 000
By trial and error, next quotient digit is 7
`300 xx 1^2 + 30 xx 1 xx 7 + 7^2=300 + 210 + 49=559`

So our new divisor is 559
 1.   
1 5.000  
1   
559 4000 `300 xx 1^2 + 30 xx 1 xx 7 + 7^2=300 + 210 + 49=559`


Step-4 :
Now multiply 559 by 7 (`559 xx 7=3913`) and subtract it.

 1.7  
1 5.000  
1   
559 4000 `300 xx 1^2 + 30 xx 1 xx 7 + 7^2=300 + 210 + 49=559`
3913  
87  


Step-5 :
Now, we have to bring down the next three digits 000
By trial and error, next quotient digit is 0
`300 xx 17^2 + 30 xx 17 xx 0 + 0^2=86700 + 0 + 0=86700`

So our new divisor is 86700
 1.7   
1 5.000000  
1    
559 4000  `300 xx 1^2 + 30 xx 1 xx 7 + 7^2=300 + 210 + 49=559`
3913   
86700 87000 `300 xx 17^2 + 30 xx 17 xx 0 + 0^2=86700 + 0 + 0=86700`


Step-6 :
Now multiply 86700 by 0 (`86700 xx 0=0`) and subtract it.

 1.70  
1 5.000000  
1    
559 4000  `300 xx 1^2 + 30 xx 1 xx 7 + 7^2=300 + 210 + 49=559`
3913   
86700 87000 `300 xx 17^2 + 30 xx 17 xx 0 + 0^2=86700 + 0 + 0=86700`
0  
87000  


Step-7 :
Now, we have to bring down the next three digits 000
By trial and error, next quotient digit is 9
`300 xx 170^2 + 30 xx 170 xx 9 + 9^2=8670000 + 45900 + 81=8715981`

So our new divisor is 8715981
 1.70   
1 5.000000000  
1     
559 4000   `300 xx 1^2 + 30 xx 1 xx 7 + 7^2=300 + 210 + 49=559`
3913    
86700 87000  `300 xx 17^2 + 30 xx 17 xx 0 + 0^2=86700 + 0 + 0=86700`
0   
8715981 87000000 `300 xx 170^2 + 30 xx 170 xx 9 + 9^2=8670000 + 45900 + 81=8715981`


Step-8 :
Now multiply 8715981 by 9 (`8715981 xx 9=78443829`) and subtract it.

 1.709  
1 5.000000000  
1     
559 4000   `300 xx 1^2 + 30 xx 1 xx 7 + 7^2=300 + 210 + 49=559`
3913    
86700 87000  `300 xx 17^2 + 30 xx 17 xx 0 + 0^2=86700 + 0 + 0=86700`
0   
8715981 87000000 `300 xx 170^2 + 30 xx 170 xx 9 + 9^2=8670000 + 45900 + 81=8715981`
78443829  
8556171  


Step-9 :
Now, we have to bring down the next three digits 000
By trial and error, next quotient digit is 9
`300 xx 1709^2 + 30 xx 1709 xx 9 + 9^2=876204300 + 461430 + 81=876665811`

So our new divisor is 876665811
 1.709   
1 5.000000000000  
1      
559 4000    `300 xx 1^2 + 30 xx 1 xx 7 + 7^2=300 + 210 + 49=559`
3913     
86700 87000   `300 xx 17^2 + 30 xx 17 xx 0 + 0^2=86700 + 0 + 0=86700`
0    
8715981 87000000  `300 xx 170^2 + 30 xx 170 xx 9 + 9^2=8670000 + 45900 + 81=8715981`
78443829   
876665811 8556171000 `300 xx 1709^2 + 30 xx 1709 xx 9 + 9^2=876204300 + 461430 + 81=876665811`


Step-10 :
Now multiply 876665811 by 9 (`876665811 xx 9=7889992299`) and subtract it.

 1.7099  
1 5.000000000000  
1      
559 4000    `300 xx 1^2 + 30 xx 1 xx 7 + 7^2=300 + 210 + 49=559`
3913     
86700 87000   `300 xx 17^2 + 30 xx 17 xx 0 + 0^2=86700 + 0 + 0=86700`
0    
8715981 87000000  `300 xx 170^2 + 30 xx 170 xx 9 + 9^2=8670000 + 45900 + 81=8715981`
78443829   
876665811 8556171000 `300 xx 1709^2 + 30 xx 1709 xx 9 + 9^2=876204300 + 461430 + 81=876665811`
7889992299  
666178701  
4. Find Cube root of 1001 using long division method

Solution:
Solution
 10.0033  
1 1001.000000000000  
1       
300 1     `300 xx 1^2 + 30 xx 1 xx 0 + 0^2=300 + 0 + 0=300`
0      
30000 1000    `300 xx 10^2 + 30 xx 10 xx 0 + 0^2=30000 + 0 + 0=30000`
0     
3000000 1000000   `300 xx 100^2 + 30 xx 100 xx 0 + 0^2=3000000 + 0 + 0=3000000`
0    
300090009 1000000000  `300 xx 1000^2 + 30 xx 1000 xx 3 + 3^2=300000000 + 90000 + 9=300090009`
900270027   
30018902979 99729973000 `300 xx 10003^2 + 30 xx 10003 xx 3 + 3^2=30018002700 + 900270 + 9=30018902979`
90056708937  
--- 9673264063  

 
Number = 1001.000000000000
Cube Root = 10.0033


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.
     
1001  

Step-2 :
Now leftmost digits is 1. Now find the largest number whose cube is `<=` 1

It is 1, whose cube is 1. Write 1 as quotient and subtract 1
 1   
1 1001  
1   
0   

Step-3 :
Now, we have to bring down the next three digits 001
By trial and error, next quotient digit is 0
`300 xx 1^2 + 30 xx 1 xx 0 + 0^2=300 + 0 + 0=300`

So our new divisor is 300
 1   
1 1001  
1   
300 1 `300 xx 1^2 + 30 xx 1 xx 0 + 0^2=300 + 0 + 0=300`


Step-4 :
Now multiply 300 by 0 (`300 xx 0=0`) and subtract it.

 10  
1 1001  
1   
300 1 `300 xx 1^2 + 30 xx 1 xx 0 + 0^2=300 + 0 + 0=300`
0  
1  


Step-5 :
Now, we have to bring down the next three digits 000
By trial and error, next quotient digit is 0
`300 xx 10^2 + 30 xx 10 xx 0 + 0^2=30000 + 0 + 0=30000`

So our new divisor is 30000
 10.   
1 1001.000  
1    
300 1  `300 xx 1^2 + 30 xx 1 xx 0 + 0^2=300 + 0 + 0=300`
0   
30000 1000 `300 xx 10^2 + 30 xx 10 xx 0 + 0^2=30000 + 0 + 0=30000`


Step-6 :
Now multiply 30000 by 0 (`30000 xx 0=0`) and subtract it.

 10.0  
1 1001.000  
1    
300 1  `300 xx 1^2 + 30 xx 1 xx 0 + 0^2=300 + 0 + 0=300`
0   
30000 1000 `300 xx 10^2 + 30 xx 10 xx 0 + 0^2=30000 + 0 + 0=30000`
0  
1000  


Step-7 :
Now, we have to bring down the next three digits 000
By trial and error, next quotient digit is 0
`300 xx 100^2 + 30 xx 100 xx 0 + 0^2=3000000 + 0 + 0=3000000`

So our new divisor is 3000000
 10.0   
1 1001.000000  
1     
300 1   `300 xx 1^2 + 30 xx 1 xx 0 + 0^2=300 + 0 + 0=300`
0    
30000 1000  `300 xx 10^2 + 30 xx 10 xx 0 + 0^2=30000 + 0 + 0=30000`
0   
3000000 1000000 `300 xx 100^2 + 30 xx 100 xx 0 + 0^2=3000000 + 0 + 0=3000000`


Step-8 :
Now multiply 3000000 by 0 (`3000000 xx 0=0`) and subtract it.

 10.00  
1 1001.000000  
1     
300 1   `300 xx 1^2 + 30 xx 1 xx 0 + 0^2=300 + 0 + 0=300`
0    
30000 1000  `300 xx 10^2 + 30 xx 10 xx 0 + 0^2=30000 + 0 + 0=30000`
0   
3000000 1000000 `300 xx 100^2 + 30 xx 100 xx 0 + 0^2=3000000 + 0 + 0=3000000`
0  
1000000  


Step-9 :
Now, we have to bring down the next three digits 000
By trial and error, next quotient digit is 3
`300 xx 1000^2 + 30 xx 1000 xx 3 + 3^2=300000000 + 90000 + 9=300090009`

So our new divisor is 300090009
 10.00   
1 1001.000000000  
1      
300 1    `300 xx 1^2 + 30 xx 1 xx 0 + 0^2=300 + 0 + 0=300`
0     
30000 1000   `300 xx 10^2 + 30 xx 10 xx 0 + 0^2=30000 + 0 + 0=30000`
0    
3000000 1000000  `300 xx 100^2 + 30 xx 100 xx 0 + 0^2=3000000 + 0 + 0=3000000`
0   
300090009 1000000000 `300 xx 1000^2 + 30 xx 1000 xx 3 + 3^2=300000000 + 90000 + 9=300090009`


Step-10 :
Now multiply 300090009 by 3 (`300090009 xx 3=900270027`) and subtract it.

 10.003  
1 1001.000000000  
1      
300 1    `300 xx 1^2 + 30 xx 1 xx 0 + 0^2=300 + 0 + 0=300`
0     
30000 1000   `300 xx 10^2 + 30 xx 10 xx 0 + 0^2=30000 + 0 + 0=30000`
0    
3000000 1000000  `300 xx 100^2 + 30 xx 100 xx 0 + 0^2=3000000 + 0 + 0=3000000`
0   
300090009 1000000000 `300 xx 1000^2 + 30 xx 1000 xx 3 + 3^2=300000000 + 90000 + 9=300090009`
900270027  
99729973  


Step-11 :
Now, we have to bring down the next three digits 000
By trial and error, next quotient digit is 3
`300 xx 10003^2 + 30 xx 10003 xx 3 + 3^2=30018002700 + 900270 + 9=30018902979`

So our new divisor is 30018902979
 10.003   
1 1001.000000000000  
1       
300 1     `300 xx 1^2 + 30 xx 1 xx 0 + 0^2=300 + 0 + 0=300`
0      
30000 1000    `300 xx 10^2 + 30 xx 10 xx 0 + 0^2=30000 + 0 + 0=30000`
0     
3000000 1000000   `300 xx 100^2 + 30 xx 100 xx 0 + 0^2=3000000 + 0 + 0=3000000`
0    
300090009 1000000000  `300 xx 1000^2 + 30 xx 1000 xx 3 + 3^2=300000000 + 90000 + 9=300090009`
900270027   
30018902979 99729973000 `300 xx 10003^2 + 30 xx 10003 xx 3 + 3^2=30018002700 + 900270 + 9=30018902979`


Step-12 :
Now multiply 30018902979 by 3 (`30018902979 xx 3=90056708937`) and subtract it.

 10.0033  
1 1001.000000000000  
1       
300 1     `300 xx 1^2 + 30 xx 1 xx 0 + 0^2=300 + 0 + 0=300`
0      
30000 1000    `300 xx 10^2 + 30 xx 10 xx 0 + 0^2=30000 + 0 + 0=30000`
0     
3000000 1000000   `300 xx 100^2 + 30 xx 100 xx 0 + 0^2=3000000 + 0 + 0=3000000`
0    
300090009 1000000000  `300 xx 1000^2 + 30 xx 1000 xx 3 + 3^2=300000000 + 90000 + 9=300090009`
900270027   
30018902979 99729973000 `300 xx 10003^2 + 30 xx 10003 xx 3 + 3^2=30018002700 + 900270 + 9=30018902979`
90056708937  
9673264063  



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7. Square root by long division method
(Previous method)		9. Babylonian method for Square root
(Next method)


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