# BODMAS - Rules for Simplifying an Expression

## Notes

### What exactly is BODMAS?

BODMAS is a set of rules or an order for performing an arithmetic expression in order to make evaluation easier. Mathematics is all about logic, and certain rules must be followed at all times. BODMAS is one of them, and if it is not followed, the entire answer can go wrong, resulting in unnecessary marks loss.

BODMAS can also be defined as standard rules for simplifying expressions containing multiple operators.

Numbers and operators are the two main components of mathematical expressions:

• Numbers: Numbers are the values used in calculations and to represent quantities. Natural numbers, whole numbers, integers, rational numbers, irrational numbers, real numbers, and complex numbers are all types of numbers.

• Operators: The Operators are two characters combined to form an expression or equation. Addition, multiplication, division, and subtraction are the most common. When there is only one operator in an expression, solving it is simple, but when there are multiple operators, it becomes more difficult.

According to BODMAS, when solving an expression, we must first solve it with brackets, then with exponents, division, multiplication, addition, and subtraction. While solving the equations, the order must be remembered. You will get the wrong answer if you do not follow this rule.

## BODMAS Rules for Simplifying an Expression:

Mathematics is a logic-based discipline. As so often, there are some simple rules to follow that help you work out the order in which to do the calculation. These are known as the ‘Order of Operations’.

BODMAS is a useful acronym that tells you the order in which you solve mathematical problems. It's important that you follow the rules of BODMAS because without it your answers can be wrong.

The BODMAS acronym is for

• Brackets (parts of a calculation inside brackets always come first).

• Orders (numbers involving powers or square roots).

• Division.

• Multiplication.

• Subtraction.

## Numeric Expressions: BODMAS

Order of operations in Numeric Expressions

To evaluate: 8 × (5 + 3)2 ÷ 16 + 4

= 8 × (5 + 3)2 ÷ 16 + 4...….{Solve everything which is inside the brackets}

= 8 × (8)2 ÷ 16 + 4...….{Power}

= 8 × 64 ÷ 16 + 4...….{Divide}

= 8 × 4 + 4...….{Multiply}

= 36.

Remember:
Brackets may be used more than once to clearly specify the order of the operations. Different kinds of brackets, such as round brackets ( ), square brackets [ ], curly brackets { }, may be used for this purpose. When solving brackets, solve the innermost bracket first and follow it up by solving the brackets outside in turn.

## Example

Solve: 2 × {25 × [(113 - 9) + (4 ÷ 2 × 13)]}

2 × {25 × [(113 - 9) + (4 ÷ 2 × 13)]}
= 2 × {25 × [104 + (4 ÷ 2 × 13)]}
= 2 × {25 × [104 + (2 × 13)]}
= 2 × {25 × [104 + 26]}
= 2 × {25 × 130}
= 2 × 3250
= 6500.

## How to Apply BODMAS?

The BODMAS rule can be used when an expression contains multiple operators. In that case, we first simplify the brackets from the inside to the outside ( ), then evaluate the exponents or roots, simplify multiplication and division, and finally perform addition and subtraction operations while moving from left to right.

Let’s start simplifying expressions in the following order

1. Bracket: Calculate everything inside the bracket first.

For example:

4 × (12 − 10)

=4 × 2

=8

The correct answer is 8 by using the BODMAS rule.

1. Order of: Solve the power, square etc.

For example:

7 + 42

=7 + 16

=23

The correct answer is 23 by using the BODMAS rule.

1. Division and Multiplication: Since multiplication and division are equally important, they should be completed from left to right.

For example:

9 + 24 ÷ 3 × 4

=9 + 8 × 4

=9 + 32

=41

The correct answer is 41 by using the BODMAS rule.

1. Addition and Subtraction: Since addition and subtraction are equally important, they should be completed from left to right.

24 + (8 5)

=24 + 3

=27

The correct answer is 27 by using the BODMAS rule.

## BODMAS without Brackets

If there are no brackets, we can apply this rule to indices, then multiplication and division, and finally addition and subtraction. The first instructs you to multiply, then divide, while the second instructs you to do the opposite.

### Here is a BODMAS Example with Answer.

Q. Simplify 3 + 4 × 2 + 4 1

Ans. BODMAS says Multiplication first,

so multiply, 4 × 24 × 2

3 + 8 + 4 1

3 + 8 + 4 = 15

Now perform subtraction at last

15 1 = 14

The correct answer is 14 by using the BODMAS rule.

## BODMAS Rule Problems

Problem 1: Simplify 12 ÷ 4 × 2 + 33 (9 + 4)

Solution: Use the BODMAS Rule (left to right whichever operations come first, we will follow that).

12 ÷ 4 × 2 + 22 (9 + 4)

First, we will simplify bracket,

= 12 ÷ 4 × 2 + 22 13

Now we will simplify powers,

= 12 ÷ 4 × 2 + 4 13

Now we will divide 12 by 4,

= 3 × 2 + 4 13

Now we will multiply 3 and 2,

=6 + 4 13

Now we will add and subtract,

=10 13

=3

Problem 2: Simplify the expression by using the BODMAS rule:

(9 × 3 ÷ 9 + 1) × 3

Solution: Step 1: Using BODMAS Rule (left to right whichever operations come first we will follow that). Here, first, we simplify the bracket and inside the bracket, we will multiply first then division (we can do vice versa) and then addition. Thus, we need to multiply 9 by 3 in the given expression,

(9 × 3 ÷ 9 + 1) × 3 and we get,

(27 ÷ 9 + 1) × 3

Step 2: Now, we need to divide 27 by 9 inside the bracket, and we get, (3 + 1) × 3

Step 3: Remove the parentheses after adding 3 and 1, we get, 4 × 34 × 3

Step 4: Multiply 4 by 3 to get the final answer, which is 12.

(9 × 3 ÷ 9 + 1) × 3 = 12

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