Cube and cube roots (find)

Arithmetic :cube and cube roots

Cube: raised to power 3

When a number is raised to the power three, the result is called the cube of the number. Or in other words we take the same number three times and multiply it to get the cube. The cube of
  • 2 is 8 (2×2×2)
  • 3 is 27 (3×3×3)
  • 4 is 64 (4×4×4)
  • 5 is 125 (5×5×5)
  • 2.5 is 15.625 (2.5×2.5×2.5)
  • 1.2 is 1.728

Perfect Cube: number three times itself

When the cube of a natural number is a natural number it is called a perfect cube. The perfect cubes of
  • 2 is 8 (2×2×2)
  • 3 is 27 (3×3×3)
  • 4 is 64 (4×4×4)
  • 5 is 125 (5×5×5)
  • So, 8, 27, 64, 125 etc are perfect cubes.
  • A given number is a perfect cube, if it can be expressed as the product of exact number of triples of equal factors.
  • To check whether 216 is a perfect cube.
    1. Find the prime factors of 216.
      216 = 2×2×2×3×3×3.
    2. Make triples of them
      216 = (2×2×2)×(3×3×3).
    3. As, 216 can be expressed as exact triples of equal factors.
      ∴ 216 is a perfect cube.

Cube Root

The cube root of a number x is a number whose cube is x. The cube root of x is denoted by x.
  • 8 is 2 |(2×2×2)
  • 27 is 3 |(3×3×3)
  • 64 is 4 |(4×4×4)
  • 125 is 5 |(5×5×5)
  • 15.625 is 2.5
  • 1.728 is 1.2

Factorization method: Cube Root (Find)

The cube root of a perfect cube is found by this method. Let us take example of 216.
  • Step 1: Resolve the given number into prime factors.
      216 = 2 × 2 × 2 ×3 × 3 × 3.
  • Step 2: Make triples of equal factors.
    900 = (2 × 2 × 2) × (3 × 3 × 3).
  • Step 3: Choose one number from each triple.
    The numbers chosen are 2 and 3.
  • Step 4: Find product of all of the numbers.
    ∛216 = 2 × 3  = 6.

Algebra of cube and cube roots

Cube : variable three times itself

  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 number it is called the cube root of the number. 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.
    cube root of 2 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.

    cube root of 3 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.

Arithmetic


  1. Looking at the above two procedures to find the cube root of two and three digit numbers. 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. Examples: cube root


    12. Cube root of 15625 is 25.

      cube root of 15625


    13. Cube root of 2352637

      cube root of 2352637



    14. Cube root of 2352.637

      cube root of 2352.637



    15. Cube root of 2.352637

      cube root of 2.352637



    16. Cube root of 2 is 1.259...
      cube root of 2




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