Cube root Part 2

Arithmetic

1. Looking at the two procedures to find the cube root of two and three digit numbers in previous post (Cube roots Part 1). We can follow the following procedure to find the cube root of any number. This method is called the division method.
   
   
   1. Group the digits from right into triples. If the number has a decimal part, make triples to both sides of decimal. If the decimal part has one or two digit left, add one or two zero correspondingly to it.
      
   
   
   2. Find  a number whose  cube is less than or equal to the first triple or the remaining digits after forming triple. Take it as divisor and quotient.
      
   
   
   3. Subtract the cube of divisor  from the first triple or the remaining digits after forming triples.
   
   
   4. Bring down the next triple to the right of the remainder. This is the new dividend.
   
   
   5. Take the thrice of quotient below on the left column of the new dividend.
   
   
   6. The new divisor is obtained by annexing the thrice of the quotient by a digit. The digit is such that [(10×(the thrice of quotient)×(quotient annexed  by new digit)×(new digit)) + (new digit)³] is less than or equal to the new dividend.
   
   
   7. Annex the new digit to the top quotient.
   
   
   8. Subtract the number obtained by [(10×(the thrice of quotient)×(quotient annexed  by new digit)×(new digit)) + (new digit)³]  from the dividend.
   
   
   9. Repeat the process.
   
   
   10. In case of taking quotient to decimal, add zeros to right of remainder in triples.
   
   
   11. Cube root of 15625 is 25.
       
       
       
       
       
   
   
   12. Cube root of 2352637
       
       
       
       
       
       
   
   
   13. Cube root of 2352.637
       
       
       
       
       
       
   
   
   14. Cube root of 2.352637
       
       
       
       
       
       
   
   
   15. Cube root of 2 is 1.259...
       
       
       
       
       

Supplements

Read Basics of the above kind of mathematics.

Cube roots Part 1

Arithmetic

Cube

1. When we multiply a variable with itself twice it is called a cube of the variable. The cube of a is a³.


2. (a + b)³ = a³ + 3a²b + 3ab² + b³


3. (a + b + c)³ = a³ + b³ + c³ + 3a²b + 3ab² + 3a²c + 3ac² + 3b²c + 3bc² + 6abc


4. We can write a two digit number as (10a + b).
   
   The cube of this number is (10a + b)³ = 1000a³ + 100×3a²b + 10×3ab² + b³.
   
   It can be written as (a³)(3a²b)(3ab²)(b³).
   
   Read Basics of the following kind of mathematics.
   
   Suppose we want to find the cube of 13 then 13³ is (1)(9)(27)(27)
   or (1)(9)(29)(7)
   or (1)(11)(9)(7)
   or (2)(1)(9)(7)
   or 2197
   and cube of 25 is (8)(60)(150)(125)
   = (8)(60)(162)(5)
   = (8)(76)(2)(5)
   = (15)(6)(2)(5)
   = 15625.


5. We can write a three digit number as (100a + 10b + c).
   
   The cube of this number is (100a + 10b + c)³ = 1000000a³ + 100000×3a²b + 10000×3ab² + 10000×3a²c + 1000×b³ + 1000×6abc + 100×3ac² + 100×3b²c + 10×3bc² + c³
   
   It can be written as (a³) (3a²b) (3ab² + 3a²c) (b³ + 6abc) (3ac² + 3b²c) (3bc²) (c³)
   
   Suppose we want to find the cube of 133 then 133³ is
   (1)(9)(36)(81)(108)(81)(27)
   or (1)(9)(36)(81)(108)(83)(7)
   or (1)(9)(36)(81)(116)(3)(7)
   or (1)(9)(36)(92)(6)(3)(7)
   or (1)(9)(45)(2)(6)(3)(7)
   or (1)(13)(5)(2)(6)(3)(7)
   or (2)(3)(5)(2)(6)(3)(7)
   or 2352637
   
   and cube of 255 is (8)(60)(210)(425)(525)(375)(125)
   = (8)(60)(210)(425)(525)(387)(5)
   = (8)(60)(210)(425)(563)(7)(5)
   = (8)(60)(210)(481)(3)(7)(5)
   = (8)(60)(258)(1)(3)(7)(5)
   = (8)(85)(8)(1)(3)(7)(5)
   = (16)(5)(8)(1)(3)(7)(5)
   = 16581375


6. We can write a three digit number with decimal point before the last digit as (10a + b + 10^-1c).
   
   The cube of this number is (10a + b + 10^-1c)³ = 1000a³ + 100×3a²b + 10×3ab² + 10×3a²c + b³ + 6abc + 10^-1×3ac² + 10^-1×3b²c + 10^-2×3bc² + 10^-3c³
   
   
   It can be written as (a³) (3a²b) (3ab² + 3a²c) (b³ + 6abc). (3ac² + 3b²c) (3bc²) (c³).
   
   Suppose we want to find the cube of 13.3 then 13.3³ is
   (1)(9)(36)(81).(108)(81)(27)
   or (1)(9)(36)(81).(108)(83)(7)
   or (1)(9)(36)(81).(116)(3)(7)
   or (1)(9)(36)(92).(6)(3)(7)
   or (1)(9)(45)(2).(6)(3)(7)
   or (1)(13)(5)(2).(6)(3)(7)
   or (2)(3)(5)(2).(6)(3)(7)
   or 2352.637
   
   and cube of 2.55 is (8)(60)(210).(425)(525)(375)(125)
   = (8)(60)(210).(425)(525)(387)(5)
   = (8)(60)(210).(425)(563)(7)(5)
   = (8)(60)(210).(481)(3)(7)(5)
   = (8)(60)(258).(1)(3)(7)(5)
   = (8)(85)(8).(1)(3)(7)(5)
   = (16)(5)(8).(1)(3)(7)(5)
   = 1658.1375

Cube Root

1. The number which when multiplied by itself twice gives the cube of the number, the number is called the cube root of the cube. The cube root of a³ is a.


2. The cube root of a³ + 3a²b + 3ab² + b³ is (a+b).


3. The cube root of a³ + b³ + c³ + 3a²b + 3ab² + 3a²c + 3ac² + 3b²c + 3bc² + 6abc is (a + b + c).


4. We can represent a two digit number as (10a + b)or(a)(b). The cube of it is (1000a³ + 100×3a²b + 10×3ab² + b³) or (a³)(3a²b)(3ab²)(b³). The figure describes how to find the cube root of a number whose cube root is a two digit number.
   
   
   
   
   1. Take cube of a, (a³) and subtract it from the number left after making group of three from right.
   2. Bring (3a) below.
   3. Suffix (b) to it.
      
   4. Main step: Multiply ten times of (3a) in column 1 row 2 ((3a) from (3a)(b)) leaving the new digit (b) to the number (a)(b) in the column 2 row 1 and new digit (b) to get [(3a²b)(3ab²)(0)], add the cube of new number (b) to it to get (3a²b)(3ab²)(b³). The expression looks like this [(3a)(0) ×(a)(b)×(b) + (b³)].
   5. For detailed method read the points in the topic below as arithmetic.
      


5. We can represent a two digit number as (100a + 10b + c) or (a)(b)(c). The cube of it is 1000000a³ + 100000×3a²b + 10000×3ab² + 10000×3a²c + 1000×b³ + 1000×6abc + 100×3ac² + 100×3b²c + 10×3bc² + c³ or (a³) (3a²b) (3ab² + 3a²c) (b³ + 6abc) (3ac² + 3b²c) (3bc²) (c³). The figure describes how to find the cube root of a number whose cube root is a three digit number.
   
   
   
   
   
   
   1. Take cube of a and subtract it from the digit left after making triples.
   2. Bring (3a) below in column 1.
      
   3. Suffix (b) to (3a) to get (3a)(b). Bring next triad below.
      
   4. Main step: Multiply ten times of (3a) in column 1 row 2 ((3a) from (3a)(b)) leaving the new digit (b) to the number (a)(b) in the column 2 row 1 and new digit (b) to get [(3a²b)(3ab²)(0)] add the cube of new number (b³) to it to get (3a²b)(3ab²)(b³). The expression looks like this [(3a)(0)×(a)(b)×(b)+ (b³)].
   5. Subtract and bring the next triad (a group of three digits) below.
   6. Suffix (3b) to (3a) to get (3a)(3b).
      
   7. Main step: Multiply ten times of  (3a)(3b) in column 1 row 2 ((3a)(3b) from (3a)(3b)(c)) leaving the new digit (c) to the number (a)(b)(c) in the column 2 row 1 and new digit (c) to get [(3a²c)(6abc)(3ac²+3b²c)(3bc²)(0)] add the cube of new number (c³) to it to get [(3a²c)(6abc)(3ac²+3b²c)(3bc²)(c³)]. The expression looks like this [(3a)(3b)(0)×(a)(b)(c)×(c)+ (c³)].
   8. Subtract this from above.
   9. For detailed method read the points in the topic below as arithmetic.