## Arithmetic

### Cube

- When we multiply a variable with itself twice it is called a cube of the variable. The cube of a is a
^{3}. - (a + b)
^{3}= a^{3}+ 3a^{2}b + 3ab^{2}+ b^{3} - (a + b + c)
^{3}= a^{3}+ b^{3}+ c^{3}+ 3a^{2}b + 3ab^{2}+ 3a^{2}c + 3ac^{2}+ 3b^{2}c + 3bc^{2}+ 6abc - We can write a two digit number as (10a + b).

The cube of this number is (10a + b)^{3}= 1000a^{3}+ 100×3a^{2}b + 10×3ab^{2}+ b^{3}.

It can be written as (a^{3})(3a^{2}b)(3ab^{2})(b^{3}).

Read Basics of the following kind of mathematics.

Suppose we want to find the cube of 13 then 13^{3}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. - We can write a three digit number as (100a + 10b + c).

The cube of this number is (100a + 10b + c)^{3}= 1000000a^{3}+ 100000×3a^{2}b + 10000×3ab^{2}+ 10000×3a^{2}c + 1000×b^{3}+ 1000×6abc + 100×3ac^{2}+ 100×3b^{2}c + 10×3bc^{2}+ c^{3}

It can be written as (a^{3}) (3a^{2}b) (3ab^{2}+ 3a^{2}c) (b^{3}+ 6abc) (3ac^{2}+ 3b^{2}c) (3bc^{2}) (c^{3})

Suppose we want to find the cube of 133 then 133^{3}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 - We can write a three digit number with decimal point before the last digit as (10a + b + 10
^{-1}c).

The cube of this number is (10a + b + 10^{-1}c)^{3}= 1000a^{3}+ 100×3a^{2}b + 10×3ab^{2}+ 10×3a^{2}c + b^{3}+ 6abc + 10^{-1}×3ac^{2}+ 10^{-1}×3b^{2}c + 10^{-2}×3bc^{2}+ 10^{-3}c^{3}

It can be written as (a^{3}) (3a^{2}b) (3ab^{2}+ 3a^{2}c) (b^{3}+ 6abc). (3ac^{2}+ 3b^{2}c) (3bc^{2}) (c^{3}).

Suppose we want to find the cube of 13.3 then 13.3^{3}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

- 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 a
^{3}is a. - The cube root of a
^{3}+ 3a^{2}b + 3ab^{2}+ b^{3}is (a+b). - The cube root of a
^{3}+ b^{3}+ c^{3}+ 3a^{2}b + 3ab^{2}+ 3a^{2}c + 3ac^{2}+ 3b^{2}c + 3bc^{2}+ 6abc is (a + b + c). - We can represent a two digit number as (10a + b)or(a)(b). The cube of it is (1000a
^{3}+ 100×3a^{2}b + 10×3ab^{2}+ b^{3}) or (a^{3})(3a^{2}b)(3ab^{2})(b^{3}). The figure describes how to find the cube root of a number whose cube root is a two digit number.

- Take cube of a, (a
^{3}) and subtract it from the number left after making group of three from right. - Bring (3a) below.
- Suffix (b) to it.

- 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 [(3a
^{2}b)(3ab^{2})(0)], add the cube of new number (b) to it to get (3a^{2}b)(3ab^{2})(b^{3}). The expression looks like this [(3a)(0) ×(a)(b)×(b) + (b^{3})]. - For detailed method read the points in the topic below as arithmetic.

- Take cube of a, (a
- We can represent a two digit number as (100a + 10b + c) or (a)(b)(c). The cube of it is 1000000a
^{3}+ 100000×3a^{2}b + 10000×3ab^{2}+ 10000×3a^{2}c + 1000×b^{3}+ 1000×6abc + 100×3ac^{2}+ 100×3b^{2}c + 10×3bc^{2}+ c^{3}or (a^{3}) (3a^{2}b) (3ab^{2}+ 3a^{2}c) (b^{3}+ 6abc) (3ac^{2}+ 3b^{2}c) (3bc^{2}) (c^{3}). The figure describes how to find the cube root of a number whose cube root is a three digit number.

- Take cube of a and subtract it from the digit left after making triples.
- Bring (3a) below in column 1.

- Suffix (b) to (3a) to get (3a)(b). Bring next triad below.

- 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 [(3a
^{2}b)(3ab^{2})(0)] add the cube of new number (b^{3}) to it to get (3a^{2}b)(3ab^{2})(b^{3}). The expression looks like this [(3a)(0)×(a)(b)×(b)+ (b^{3})]. - Subtract and bring the next triad (a group of three digits) below.
- Suffix (3b) to (3a) to get (3a)(3b).

- 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 [(3a
^{2}c)(6abc)(3ac^{2}+3b^{2}c)(3bc^{2})(0)] add the cube of new number (c^{3}) to it to get [(3a^{2}c)(6abc)(3ac^{2}+3b^{2}c)(3bc^{2})(c^{3})]. The expression looks like this [(3a)(3b)(0)×(a)(b)(c)×(c)+ (c^{3})]. - Subtract this from above.
- For detailed method read the points in the topic below as arithmetic.

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