Simplification 

When the express the given expression in the simplest form, then it is known as simplification. In order to simplify an expression, we use the operations in the following order, which is known as BODMAS rule.

Rule of BODMAS

• B - Bracket


• O - Of


• D - Division 


• M - Multiplication


• A - Addition


• S - Substraction

Order of Brackets

1. Vi - Viniculum of Bar '-'


2. Ci - Circular Bracket ()


3. Cu - Curly Bracket {}


4. Sq - Square Bracket []

IMPORTANT FORMULA

1.  a÷b÷c=a×1b×1c
2. abc+ def - ghi = (a+d-g)+bc+ef-hi
3. (a+b)2-(a-b)2=4ab
4. x+x+x+....∞=n+1, if x=n(n+1)
5. m of n%=m×n100
6. a-m=1am
7. am+1=am.a
8. am×a-n=am-n
9. am-1=am×1a, amn×nm=a, ax=1, then x=0

Simple and Mixed Fractions

Fractions Those numbers which can be expressed as in the forn of pq where q≠0, are known as fractions, where p is known as numaerator of fraction and q is known as denominator of fraction.
Mixed Fractions Those fractions which consists a whole number and proper fraction, are known as mixed fractions. e.g., 314  is a mixed fraction.

IMPORTANT FORMULA

1. bc+ec=b+ec
2. bc+ef=b×f+e×cc×f
3. ab÷cd=ab×dc
4. a×bcd=a×b+a×cd
5. abc+dec=(a+d)+b+ec
6. abc+def=(a+d)+b×f+c×ec×f
7. abc=a×cb
8. ab÷bc÷ce=ab×cb×ed

Algebra

This is an important part of simplification. In which some difficult problem are solved by using the algebraic formula

IMPORTANT FORMULA

1. (a+b)2=a2+2ab+b2=(a-b)2+4ab
2. (a-b)2=a2-2ab+b2=(a+b)2-4ab
3. a2+b2=(a+b)2+(a-b)22
4. a2-b2=(a+b)(a-b)
5. (a+b)3=a3+b3+3ab(a+b)
6. (a-b)3=a3-b3-3ab(a-b)
7.  a3+b3=(a+b)(a2-ab+b2)
8. a3-b3=(a-b)(a2+ab+b2)
9. ab=(a+b)2-(a-b)24
10. (a+b+c)2=a2+b2+c2+2(ab+bc+ca)
11. a3+b3+c3-3abc=(a+b+c)(a2+b2+c2-ab-bc-ca)
12. if a+b+c=0, then a3+b3+c3=3abc

Percentage

Per Cent is made of two words per and cent. Meaning of per and cent is "per hundred". Sign of per cent is '%'. Meaning of 35% is 35100 1.e., 35 parts out of 100 parts.

IMPORTANT FORMULAS

1. n% of m=m×n100
2. Number +A% of Number=Number×100+A100
3.Number - A% of Number=Number×100-A100
 
4. If the value of number is first increased by A% and later decreased by A% then the next change is always a decrease which is equel to A2100%
5. If an object price in increased or decreased by A% and the other factor is increased or decreased by B%, then the net effect is given by
Net effect=±A±B+±A±B100%

Some Quick Results

2%=150	4%=125	8%=225	16%=425	64%=1625
5%=120	10%=110	20%=15	40%=25	70%=710
25%=14	50%=12	75%=34	250%=52	110%=1110
614%=116	1212%=18	1623%=16	3313%=13	6623%=23

Square - Square Root and Cube - Cube Root

Square, when any number is multiplied by itself, then the produce is known as square of the number.
e.g., Square of 5=52=5×5=25
Square Root, Square may be define the number whose square is equal to given number. In other words, square root of a given number is the number when multiplied by itself, gives the product equal to the given number.
e.g., Square root of 16=16=4
Cube, cube of a number is the triple product obtained on multiplying the number by itself.
e.g., Cube of 5=53=5×5×5=125
Cube Root, cube root of a number is the number whose cube is the given number.
e.g., Cube root of 216=2163=6×6×63=6

Surds and Indices

Surds, any root of rational number which cannot be easily calculated is known as Surd.

Rules of Surds

1. ann=a
2. anm=amn
3. abn=an×bn
4. anm=amn
5. abn=anbn
6. a×b×c=abc
 
Indices, Let a be a real number amd m is a positive integer, then a×a×a..... m times= am. This is read as 'a raise to power m'. Here,a is known as base and m is known as index.

Rules of Indices

1. am×an=am+n
2. am÷an=am-n
3. (am)n=amn
4. abm=am×bm
5. abm=ambm
6. a0=1

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