Order Of Operations – Definition, Rules & Examples

Order of Operations is a collection of mathematical principles that determine the order in which computations are to be executed in an expression. The order of operations and rules are expressed here:

• Brackets ( ), { }, [ ]
• Exponents
• Division (÷) and Multiplication (×)
• Addition (+) and Subtraction (-)

Here, Parentheses come first, followed by exponents, multiplication and division (from left to right), and addition and subtraction.

These guidelines guarantee that everyone gets the same solution while solving a problem.

There are multiple acronyms used to define the order of operation, such as BODMAS, BEDMAS, BIDMAS or PEDMAS .

Order of Operations Rules-PEMDAS vs BODMAS

Order of Operations principles specify the order in which mathematical equations are solved, maintaining consistency and correctness throughout calculations. These criteria are critical for preventing misunderstanding and producing accurate outcomes. They include parentheses, exponents, multiplication and division, and addition and subtraction, which are often known by acronyms like as PEMDAS or BODMAS.

PEMDAS Rule

PEMDAS is an abbreviation for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction. This rule prioritizes calculations in brackets first, then exponents, multiplication and division, and finally addition and subtraction.

Order of operation in PEDMAS are,

• P stands for Parentheses ( ), { }, [ ]
• E stands for Exponents (a^b)
• M stands for Multiplication(×)
• D stands for Division(÷)
• A stands for Addition(+)
• S stands for Subtraction(-)

Examples of PEMDAS

Let's solve the expression (3 + 2) × 4 - 6 ÷ 2 using PEMDAS

Step 1: Inside Parentheses: (3 + 2) × 4 - 6 ÷ 2 = 5 × 4 - 6 ÷ 2
Step 2: Multiplication: 5 × 4 - 6 ÷ 2 = 20 - 6 ÷ 2
Step 3: Division: 20 - 6 ÷ 2 = 20 - 3
Step 4: Subtraction: 20 - 3 = 17

So, the result is 17

BODMAS Rule

BODMAS is an abbreviation for brackets, orders (or exponents), division and multiplication (from left to right), and addition and subtraction (from left to right). BODMAS, like PEMDAS, emphasizes the significance of prioritizing computations within brackets or brackets first, followed by exponents, division and multiplication, and lastly addition and subtraction. BODMAS provides consistency and precision in mathematical computations.

Order of operation in BODMAS are:

• B stands for Brackets ( ), { }, [ ]
• O stands for Order
• D stands for Division (÷)
• M stands for Multiplication (×)
• A stands for Addition (+)
• S stands for Subtraction (-)

Examples of BODMAS:

Let's solve the expression 6 + 3 × 2 - 4 ÷ 2 using BODMAS

Step 1: Multiplication: 6 + 3 × 2 - 4 ÷ 2 = 6 + 6 - 4 ÷ 2
Step 2: Division: 6 + 6 - 4 ÷ 2 = 6 + 6 - 2
Step 3: Addition: 6 + 6 - 2 = 12 - 2
Step 4: Subtraction: 12 - 2 = 10

So, the result is 10

How to Use Order of Operations?

To utilize the Order of Operations, go in the following order: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction. Begin with completing operations within brackets, then assess exponents, multiplication and division, and ultimately addition and subtraction.

Order of Operations with Parentheses

First, solve any calculations in brackets. The remainder of the phrase should then be evaluated in accordance with the Order of Operations.

Example:Simplify 2 × (3+4)

Step 1: Solve inside the parentheses: 2 × (3+4) = 2 × 7
Step 2: Multiply: 2 × 7 = 14

Following the Order of Operations, the equation equals 14

Order of Operations with Exponents

After the brackets, address any exponents or powers in the phrase. Compute these before proceeding to the remaining operations.

Example:Simplify 2³ × 4

Step 1: Evaluate the exponent: 2³ × 4 = 8 x 4
Step 2: Multiply: 8 x 4 = 32

Following the Order of Operations, the equation equals 32

Order of Operations with Multiplication or Division and Addition or Subtraction

After working with parentheses and exponents, execute multiplication and division from left to right, followed by addition and subtraction from left to right.

Example:Simplify 6 + 4 × 3 - 2

Step 1: Multiply: 6 + 4 × 3 - 2 = 6 + 12 − 2
Step 2: Add: 6 + 12 - 2 = 18 − 2
Step 3: Subtract: 18 - 2 = 16

Following the Order of Operations, the equation equals 16.

Solved Examples on Order of Operations

Example 1: Solve expression: 8 + (5 × 3) − 2² using PEMDAS.

Solution:

Step 1: Parentheses: 8 + (5 × 3) − 2² = 8 + (15) − 2²
Step 2: Exponents: 8 + (15) − 2² = 8 + 15 − 4
Step 3: Addition: 8 + 15 − 4 = 23 − 4
Step 4: Subtraction: 23 − 4 = 19

Therefore, the solution is 19

Example 2: Solve expression: 12 − 4 × (6 ÷ 2) + 5 using BODMAS.

Solution:

Step 1: Brackets: 12 − 4 × (6 ÷ 2) + 5 = 12 − 4 × 3 + 5
Step 2: Multiplication: 12 − 4 × 3 + 5 = 12 − 12 + 5
Step 3: Addition: 12 − 12 + 5 = 12 - 17
Step 4: Subtraction: 12 - 17 = 5

Therefore, the solution is 5

Example 3: Solve expression: 3 × (4 + 2)² − 10 using Order of operation.

Solution:

Step 1: Parentheses: 3 × (4 + 2)² − 10 = 3 × (6)² − 10
Step 2: Exponents: 3 × (6)² − 10 = 3 × 36− 10
Step 3: Multiplication: 108− 10
Step 4: Subtraction: 98

Therefore, the solution is 98