Simplification Formulas
Dear Aspirants, This page will tell you about Important Simplification Formulas and Concepts as well as order of operations(i.e. BODMAS Rule).
What is BODMAS Rule ?
BODMAS rule defines the correct sequence in which operations are to be performed in a given mathematical expression to find its value.
In BODMAS,
B = Bracket
O = Order (Powers, Square Roots, etc.)
DM = Division and Multiplication (left-to-right)
AS = Addition and Subtraction (left-to-right)
In some countries, the acronym PEMDAS is used instead of BODMAS. PEMDAS stands for “Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction”. Some other variations used to represent the same concept are BIDMAS, ERDMAS, PERDMAS and BPODMAS.
Order of Operations :
- While simplifying an expression, the following order must be followed.
- Do operations in brackets first, strictly in the order (), {} and []
- Evaluate exponents (powers, roots, etc.)
- Perform division and multiplication, working from left to right. (division and multiplication rank equally and done left to right).
- Perform addition and subtraction, working from left to right. (addition and subtraction rank equally and done left to right).
Examples :
(a) 12+22÷11×(18÷3)2−1012+22÷11×(18÷3)2−10
=12+22÷11×62−10 =12+22÷11×62−10 (∵ brackets first)
=12+22÷11×36−10 =12+22÷11×36−10 (∵ exponents)
=12+2×36−10 =12+2×36−10 (∵ division and multiplication, left to right)
=12+72−10 =12+72−10 (∵ division and multiplication, left to right)
=84−10 =84−10 (∵ addition and subtraction, left to right)
=74=74
(b) 4+10−3×6/3+44+10−3×6/3+4
=4+10−18/3+44+10−18/3+4 (∵ division and multiplication, left to right)
=4+10−6+44+10−6+4 (∵ division and multiplication, left to right)
=14−6+44+10−6+4 (∵ addition and subtraction, left to right)
=8+54-2 =8+52 (∵ addition and subtraction, left to right)
=60
(c) 1+2/2×21+2/2×2
=1+1×21+1×2 (∵ division and multiplication, left to right)
=1+21+2 (∵ division and multiplication, left to right)
=24
Modulus of a Real Number:
Modulus of a real number a is defined as
| |a| = | a, if a > 0 |
| –a, if a < 0 |
Thus, |5| = 5 and |-5| = -(-5) = 5.
Virnaculum (or Bar):
When an expression contains Virnaculum, before applying the ‘BODMAS’ rule, we simplify the expression under the Virnaculum.
In VBODMAS, V : Virnaculum or bar B : Bracket O : Of or Order D : Division M : Multiplication A : Addition S : Subtraction
Modulus Modulus of a real number x is its positive value, denoted by |x|. Thus, |7| = 7 and |−7| = 7
Formulas :
(a + b)2 = a2 + b2 + 2ab
(a – b)2 = a2 + b2 – 2ab
(a + b) (a – b) = a2 – b2
(a + b)2 – (a – b)2 = 4ab
(a + b)2 + (a – b)2 = 2(a2 + b2)
(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ac)
(a3 + b3) = (a + b)(a2 – ab + b2)
(a3 – b3) = (a – b)(a2 + ab + b2)
(a + b)3 = a3 + b3 + 3ab(a + b)
(a – b)3 = a3 – b3 – 3ab(a – b)
(a3 + b3 + c3 – 3abc) = (a + b + c) (a2 + b2 + c2 – (ab + bc + ca)
If a + b + c = 0, then a3 + b3 +c3 = 3abc
Thank you and All the best!!
For always keep in touch with updates you can join or visit at Facebook Page or Twitter