## 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.

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