## Simplification

A process which convert complex arithmetical expression into simple expression is called simplification.

Order of Operations
When we simplify expression and equation we used order of operation.

### VBODMAS Rule

In this rule, we use operations in order
V - Vinculum or Bar
B - Brackets - Order of brackets
First- Small brackets ( )
second- Middle brackets{ }
Third- Square brackets[ ]
O - Of
D - Division
M - Multiplication A - Addition
S - Subtraction

Order of above mentioned operations is same as the order of letters in the VBODMAS from left to right

Example- Simplify 5 - [ 4 - {9 - ( 7 - 5 - 3 )}]
Solution:-
5 - [ 4 - {9 - ( 7 - 5 - 3 )}]
= 5 - [ 4 - { 9 - ( 7 - 5 + 3 )}]
= 5 - [ 4 - { 9 - ( 7 - 2 )}]
= 5 - [ 4 - { 9 - 5 }]

= 5 - [ 4 - 4 ]
= 5 - 0
= 5
Example-
Simplify 34500 ÷ 30 ÷ 10
Solution:- 34500 ÷ 30 ÷ 10
= 34500/30 ÷ 10
= 1150 ÷ 10
= 1150/10
= 115
Example

 = - 5 1 + 3 2 + 2 3 7 7 7

Solution:-
 = 5 1 + 3 2 + 2 3 7 7 7
 = (5 + 2 + 2 ) + 1 + 2 + 3 7 7 7
 = 10 5 7
 = 10 5 7

Example- Simplify 34500 ÷ 30 ÷ 10
Solution:- 34500 ÷ 30 ÷ 10
= 34500/30 ÷ 10
= 1150 ÷ 10
= 1150/10
= 115
Example
 = - 5 1 + 3 2 + 2 1 7 7 7

Solution:-
 = 5 1 + 3 2 + 2 1 7 7 7
 = (5 + 3 + 2) + 1 + 2 + 1 7 7 7
 = 10 5 7

## Basic Formulae

 (a + b)² = a² + b² + 2ab

Example- Find the square of 107.
Solution:-

( 107 )2 = ( 100 + 7 )2
= 1002 + 72 + 2 × 100 × 7
= 10000 + 49 + 1400
= 11449

 (a - b)² = a² + b² - 2ab

Example- Find the square 999.
Solution:-

( 999 )2 = ( 1000 - 1 )2
= 10002 + 12 - 2 × 1000 × 1
= 1000000 + 1 - 2000
= 998001

 a² - b² = (a + b)(a - b)

Example- Find the product of 998 and 1002.
Solution:- ( 1000 - 2 ) ( 1000 + 2 )
= 1000² - 2²
= 1000000 - 4
= 999996
Example- Find the value of 48² - 46²
Solution:- 48² - 46²
= ( 48 + 46 ) ( 48 - 46 )
= 94 × 2
= 188

 (a + b)³ = a³ + b³ + 3ab (a - b)

Example- Find the value of ( 103 )³
Solution:-

( 103 )³
= ( 100 + 3 )³
= 100³ + 3³ + 3 × 100 × 3 ( 100 + 3 )
= 1000000 + 27 + 900 × 103
= 1000000 + 27 + 92700
= 109272

 (a - b)³ = a³ - b³ - 3ab (a - b)

Example- Find the value of ( 98 )³.
Solution:- ( 98 )³
= ( 100 - 2 )³
= 100³ - 2³ - 3 × 100 × 2 ( 100 - 2 )
= 1000000 - 8 - 600 × 98
= 1000000 - 8 - 58800
= 1000000 - 58808
= 941192

 a³ + b³ = (a + b) - (a² - ab + b²)

Example- Find the value of ( 4.8 )³ + ( 1.2 )³ / ( 4.8 )² - 4.8 × 1.2 + ( 1.2 )²
Solution:- Let a = 4.8 and b = 1.2
a³ + b³ / (a² - ab + b² )
= {( a + b ) ( a2 - ab + b2 ) }/ (a2 - ab + b2)
= ( a + b )
= 4.8 + 1.2
=6

7.

 a³ - b³ = (a - b) - (a² + ab + b²)

Example- Find the value of ( 11.3 )³ -( 1.3 )³ / ( 11.3 )³ +11.3 ×1.3 +( 1.3 )³
Solution:- Let a = 11.3 and b = 1.3
= a³ - b³ / a² + ab + b²
= ( a - b ) ( a² + ab + b² ) / a² + ab + b²
= ( a - b )
= 11.3 - 1.3
=10

8.

 (a + b + c)² = a² + b² + C² +2ab + 2bc + 2ca

Example- If a + b + c = 14 and a² + b² + c² = 96, then find the value of ab + bc + ca .
Solution:- a + b + c = 14
Squaring on both sides
or, ( a + b + c )² = (14 )²
or, a² + b² + c² + 2ab + 2bc + 2ca = 196
or, 96 + 2ab + 2bc + 2ca = 196
or, 96 + 2 ( ab + bc + ca ) = 196
or, 2 ( ab + bc + ca )
= 196-96
or, 2 ( ab + bc + ca ) = 100
∴ ab + bc + ca
= 100/2 = 50

9.

 (a + b)² + (a - b)² = a² + b² - 2ba

= 2a² + 2b²
= 2 ( a² + b²)
∴ ( a + b )² + ( a - b )² = 2 ( a² + b²)
Example-
 Find the value of (745 + 123)² + (745 - 123)² 745 × 745 + 123 × 123

Solution:- Let a = 745 and b = 123
 = (a + b)² + (a - b)² a² + b²
 = 2 (a² + b²) a² + b²

= 2

10.

 (a + b)² - (a - b)² = a² + b² - 2ba - (a² + b² - ab)

= a² + b² + 2ab - a² - b² + 2ab

=2ab+2ab = 4ab
∴ ( a + b )² - ( a - b )² = 4ab

11.

 a³ + b³ c³ - 3abc = (a + b + c) = (a² + b² + c² - ab - bc - ca)

 = 1 (a + b + c)[( a - b )² + ( b - c )² + ( c - a )² ] 2

If a + b + c = 0, then
a3 + b3 + c3 - 3abc = 0
or, a3 + b3 + c3 = 3abc
Example- Find the value of a3 + b3 + c3 - 3abc when a = 23, b = 24 and c = 25
Solution:- a3 + b3 + c3 - 3abc = 1/2 ( a + b + c ) [ ( a - b )2 + ( b - c )2 + ( c - a )2 ]

 = 1 (23 + 24 + 25)[( 23 - 24 )² + ( 24 - 25 )² + ( 25 - 23 )²] 2
 = 1 × 72 [( -1 )2 + ( -1 )2 + ( 2 )2] 2

= 36 [ 1 + 1 + 4 ]
= 36 × 6
= 261
Example-
Solution:- Let a = 5.3, b = 3.5 and c = 1.2
 = (5.3)3 + (3.5)3 + (1.2)3 - 3 × 5.3 × 3.5 × 1.2 (5.3)2 + (3.5)2 + (1.2)2 - 5.3 × 3.5 - 3.5 × 1.2 - 1.2 × 5.3
 = a3 + b3 + c3 - 3abc a2 + b2 + c2 - ab - bc - ca

= 5.3 + 3.5 + 1.2
= 10
Example- If a = 25, b = 50 and c = -75, then find the value of a3 + b3 + c3 - 3abc.
Solution:- Here, a = 25, b = 50 and c = -75
a + b + c = 25 + 50 + ( -75 ) = 75 - 75 = 0
a3 + b3 + c3 - 3abc = ( a + b + c ) ( a2 + b2 + c2 - ab - bc - ca )
or, a3 + b3 + c3 - 3abc = 0 ( a2 + b2 + c2 - ab - bc - ca )
∴ a3 + b3 + c3 - 3abc = 0

### Some Important Rule

Rule- If x + 1/x = a, then x2 + 1/x2 = a2 - 2
( x + 1/x )2 = x2 + 1/x2 + 2 × x × 1/x
or, ( x + 1/x )2 = x2 + 1/x2 + 2
∴ x2 + 1/x2 = ( x + 1/x )2 - 2

Example- If x + 1/x = 3, then find the value x2 + 1/x2.
Solution:- x + 1/x = 3
∴ x2 + 1/x2 = ( x + 1/x )2 - 2 = 32 - 2
= 9 - 2
= 7

Example- If x + 1/x = 3, then find the value x4 + 1/x4.
Solution:- x + 1/x = 3
∴ x2 + 1/x2 = ( x + 1/x )2 - 2
= 32 - 2
= 9 - 2
= 7
∴ x4 + 1/x4 = ( x2 + 1/x2 )2 - 2
= 72 - 2
= 49 - 2
= 47

Example- If x + 1/x = 3, then find the value x8 + 1/x8.
Solution:- x + 1/x = 3
∴ x2 + 1/x2 = ( x + 1/x )2 - 2
= 32 - 2
= 9 - 2
= 7
∴ x4 + 1/x4 = ( x2 + 1/x2 )2 - 2
= 72 - 2
= 49 - 2
= 47
∴ x8 + 1/x8 = ( x4 + 1/x4 )2 - 2
= 472 - 2
= 2209 - 2
= 2207

Rule- If x + 1/x = a, then x3 + 1/x3 = a3 - 3a
( x + 1/x )3 = x3 + 1/x3 + 3 × x × 1/x ( x + 1/x )
or, ( x + 1/x )3 = x3 + 1/x3 + 3( x + 1/x )
∴ x3 + 1/x3 = ( x + 1/x )3 - 3( x + 1/x )

Example- If x + 1/x = 3, then find the value of x3 + 1/x3.
Solution:- x + 1/x = 3
∴ x3 + 1/x3 = ( x + 1/x )3 - 3( x + 1/x )
= 33 - 3( x + 1/x )
= 27 - 3 × 3
= 27 - 9
= 18

Example- If x + 1/x = 3, then find the value of x9 + 1/x9.
Solution:- x + 1/x = 3
∴ x3 + 1/x3 = ( x + 1/x )3 - 3( x + 1/x )
= 33 - 3( x + 1/x )
= 27 - 3 × 3
= 27 - 9
= 18
∴ x9 + 1/x9 = ( x3 + 1/x3 )3 - 3( x + 1/x )
= 183 - 3( x3 + 1/x3 )
= 5832 - 3 × 18
= 5832 - 54 = 5778

Rule- If x + 1/x = a, then x5 + 1/x5 = [{ ( a3 - 3a ) ( a2 - 2 ) } - a ]
x5 + 1/x5 = ( x3 + 1/x3 ) ( x2 + 1/x2 ) - ( x + 1/x )

Example- If x + 1/x = 3, then find the value of x5 + 1/x5.
Solution:- x + 1/x = 3
∴ x3 + 1/x3 = ( x + 1/x )3 - 3( x + 1/x )
= 33 - 3( x + 1/x )
= 27 - 3 × 3
= 27 - 9
= 18
∴ x2 + 1/x2 = ( x + 1/x )2 - 2
= 32 - 2
= 9 - 2
= 7
∴ x5 + 1/x5 = ( x3 + 1/x3 ) ( x2 + 1/x2 ) - ( x + 1/x )
= 18 × 7 - 3
= 126 - 3
= 123

Rule- If x + 1/x = a, then x - 1/x = √a2 - 4

Example- If x + 1/x = 5, then find the value of x - 1/x.
Solution:- x + 1/x = 5
∴ x - 1/x = √( x + 1/x )2 - 4
= √( 5 )2 - 4
= √21

Example- If x3 + 1/x3 = 10, then find the value of x3 - 1/x3.
Solution:- x3 + 1/x3 = 10
∴ x3 - 1/x3 = √( x3 + 1/x3 )2 - 4
= √( 10 )2 - 4
= √100 - 4
= √96
= 4 √6

Rule- If x - 1/x = a, then x + 1/x = √a2 + 4

Example- If x - 1/x = 7, then find the value of x + 1/x.

Solution:- x - 1/x =7

∴ x + 1/x = √( x - 1/x )2 + 4

= √( 7 )2 + 4 = √49 + 4 = √53

Example- If x = 17, then find the value of x5 - 18x4 + 18x3 - 18x2 + 18x - 19.
Solution:-
x5 - 18x4 + 18x3 - 18x2 + 18x - 19.
= x5 - 17x4 - x4 + 17x3 + x3 - 17x2 - x2 + 17x + x - 19
= 175 - 17 ( 17 )4 - 174 + 17 ( 17 )3 + 173 - 17 ( 17 )2 - 172 + 17 ( 17 ) + 17 - 19
= 175 - 175 - 174 + 174 + 173 - 173 - 172 + 172+ 17 - 19
= 17 - 19
= -2