**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÷11×6^{2}−10 =12+22÷11×6^{2}−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}** = a

^{2 }+ b

^{2 }+ 2ab

**(a – b) ^{2} **= a

^{2 }+ b

^{2 }– 2ab

**(a + b) (a – b)** = a^{2 }– b^{2}

**(a + b) ^{2 }– (a – b)^{2}** = 4ab

**(a + b) ^{2} + (a – b)^{2}** = 2(a

^{2}+ b

^{2})

**(a + b + c) ^{2} **= a

^{2}+ b

^{2}+ c

^{2}+ 2(ab + bc + ac)

**(a ^{3} + b^{3})** = (a + b)(a

^{2}– ab + b

^{2})

**(a ^{3} – b^{3})** = (a – b)(a

^{2}+ ab + b

^{2})

**(a + b) ^{3}** = a

^{3}+ b

^{3}+ 3ab(a + b)

**(a – b) ^{3}** = a

^{3}– b

^{3}– 3ab(a – b)

**(a ^{3} + b^{3} + c^{3} – 3abc)** = (a + b + c) (a

^{2}+ b

^{2}+ c

^{2}– (ab + bc + ca)

**If** **a + b + c = 0**, then **a ^{3} + b^{3} +c^{3}** = 3abc

Thank you and All the best!!

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