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Home > Pre-Algebra > Power and Radical > Cube root of a number using long division method example
13. Cube root by long division method example ( Enter your problem )
( Enter your problem )

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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