How to find cube root by division method
Answers
x*x*x = x³ _
There fore x = Cube Root of x³ = ³√x³
Please find the below table for the Cubes of 1 to 9.

Here we can notice, Cube of 3 ends with 7, 4 ends with 4, 5 ends with 5, 6 ends with 6, 7 ends with 3, 8 ends with 2 and 9 ends with 9.
Perfect Cube: 1, 8, 27, 64, 125, 216, 343, 512, 729, ..... are called Perfect Cubes whose Cube Roots are exact numbers. Any Real number multiplied itself three times, the cube is called Perfect Cube.
Easy Method for finding out Cube Roots for Perfect Cube:
Example 1: To find out Cube Root of 421875.
Solution: By grouping each 3 digits from right to left of 421875 is 421'875.
Now take the 1st group from left, which is 421 guess the which cube is maximum and less than or equal to 421. We can guess 7, as the cube of 7 is 343 will satisfy the condition. And next group from left is 875, ending with 5 means cube of 5.

So, the Cube Root is 75.
Long Division Method:
_____
To find out Cube Root of 15625: Denoted by ³√15625
Long Division Method: Set up a division with the number, with grouping each 3 digits from the decimal point to left.

Take the 1 st group from left side, which is 15 as new dividend. Check which cube is the maximum and less than equal to 15. 3³ is 27 which is more than 15. So 2³ will satisfy the condition. Now place 2 right sides, and 2³=8 below 15. Subtract 8 from 15, remainder is 7.

Bring down next group 625, and place after 7. So the new dividend is 7625. Now place _² + 30*2(10*2 _) left side as new divisor where 2 is initial quotient, blank is for new quotient and place a blank on right side after 2.

Guess a new quotient to fill in the blanks of the new divisor and quotient such that the product of new quotient and new divisor should be maximum and less than equal to 7625.
We can estimate new quotient from the formula : 380*estimate*2² = present dividend
where 2 is initial quotient
1520*estimate=7625
estimate= 7625/1520 = ~ 5
Answer:
just like in square root make pair of 3 digits from back to front. the group we get is 10 and 648
find the no. whose cube root is less than or equal to 10 . obviously its 2
subtract 8 from 10 and write down the 2nd pair i.e 648 as in square root method. we have now 2648
now we have to find the multiplication factor for further process that comes by multiplying the first obtained digit i.e 2 by 30 i.e 60
now we have to find 2nd digit lets say y
then the following behaviour should apply
(xy *multiplication factor + y^2) * y should be be the closest number than the remainder left after first process
//Note that xy is not x*y and rather a 2 digit number having x and y as its digits//
in this case by choosing 2 as “y” we get above no. 22 and adding 22*60 by 4 i.e square of 2 we get 1324 and this 1324 then multiplied by 2 i.e second digit which we chose gives 2648 leaving remainder as “0”.
So cube root of 10648 is 22
example-2
lets take cube of 21 i.e. 9261.
now make make pair of 3 from back i.e 9 and 261
find the no. whose cube root is less than or equal to 9 . obviously its 2
subtract 8 from 9 and write down the 2nd pair i.e 261 as in square root method. we have now 1261
now we have to find the multiplication factor for further process that comes by multiplying the first obtained digit i.e 2 by 30 i.e 60
now we have to find 2nd digit . the second digit is such that when the no. formed by 1 and 2nd digit multiplied by multiplication factor is added to square of the digit is multiplied by 2nd digit the result should be less than or equal to 1261.
like in this case we by choosing 1 we get above no. 21 and adding 21*60 + 1 i.e square of 1 we get 1261 and 1261 multiplied by 1 i.e 2nd digit of required no. 1261 is the result and subtracting 1261 from 1261 we get remainder 0 i.e 21 is the cube root of 9261
lets take more complex example
example 3
cube of 463 is 99252847
now first make pair of 3 from backwards i.e 99, 252, 847
64 is less than 99 so 4 is our first digit
write down 99-64= 35 suffixed with 252 i.e 35252
multiplication factor will be 30 *4=120
choosing second digit 6 gives us 46*120+36(36 being square of 6)=5556 and 5556 multiplied by 6 gives 33336....
again subtract 35252 and 33336 and suffix 3rd pair i.e 1916847
now our multiplication factor will be 46*30= 1380
choosing 3 as 3rd digit will give us 463*1380+9(square of 3) = 638949 which multiplied by 3rd digit i.e 3 gives 1916847
our remainder as we can see is zero so the cube root of 99252847 is 463
We can note that the calculation become lengthy and tedious from 3rd digit
NOW my previous answer.
short trick for finding n (cube root) when m(cube of n i.e n^3=m) is given where n is an integer less than 100
# example I - lets take 185193
we already somehow know that it is a cube of an integer. or given in question.
now make pair of 3 from back wards i.e. 185, 193
now we have to find two digits only as cube root of six digit no. cant be greater than 99
1st digit comes by taking cube of a no. less than 185 i.e 125 which is cube of 5
so our first digit is 5
now 2 nd digit comes from the unique property which states that last digit of cube of 1 to 9 is different.
0^3=0......#last digit 0
1^3=1......#last digit 1
2^3=8......#last digit 8
3^3=27....#last digit 7
4^3=64....#last digit 4
5^3=125..#last digit 5
6^3=216..#last digit 6
7^3=343..#last digit 3
8^3=512..#last digit 2
9^3=729..#last digit 9
so our 2nd pair(193) last digit is 3 which can come only by cube of any no. ending with 7 (as given earlier that no. is a perfect cube)
so our 2nd digit is 7
hence our no. is 57
# example II - now take 571787
pairs are 571, 787
8 cube =512 less than 571 so 8 our 1st digit
last digit of 2nd pair =7 which can come from only cube of 3 so 2nd digit is 3
hence the cube root of 571787 is 83