Introduction to PEMDAS and BODMAS rule explained with example

In mathematics, the BODMAS and PEMDAS rules are widely used to solve the math expression problems according to the correct order of arithmetic sequence. These rules are used to solve the problems to get the similar result all over the world.

The BODMAS rule is taught in the Asian countries and the PEMDAS rule is used in the European nation. In this post, we will study the basics of solving the problems by using the BODMAS and PEMDAS rules along with examples.

What is the order of operation?

The order of operation is a technique used to solve the problems of math expressions in the correct way. This technique is used all over the world to solve the accurate sequence of brackets, exponents, multiply, divide, addition, and subtraction.

There are two main methods used to solve the math expressions all over the world. One is the PEMDAS rule that is used all in the Western areas like the US, UK, Canada, etc.  and the other is the BODMAS rule that is used in the Asian countries like India, Pakistan, Sri Lanka, etc.

Let us discuss these rules briefly.

PEMDAS Rule

The PEMDAS rule is one of the best ways to solve the problems of math expression by solving all the mathematics symbols one by one in a correct sequence. The term PEMDAS rule stands for:

• P = Parentheses “()”
• E = Exponent “^”
• M = Multiplication “* or x”
• D = Division “/”
• A = Addition “+”
• S = Subtraction “-”

Methods to solve math expression using PEMDAS

To solve the problems of the math expression with the help of PEMDAS rule, follow the steps below.

1. Firstly, calculate the parentheses “()” of the math expression even if the parentheses are the last term of the expression.
2. Then calculate the exponent terms of the math expression if there are more than one exponent terms in the expression solve the leftmost term first.
3. After that, calculate the multiplication and division symbol from left to right present in the math expression.
4. Lastly, calculate the plus and minus symbols from left to right present in the math expression.  

Let us take an example of this rule to understand the term accurately.

Example: By PEMDAS rule

Evaluate 12 + 45 – 5³ + (18 + 2) * 3 + 10/5 – (13 + 7) + 5⁴ * 4 by using the PEMDAS rule.

Solution

Step-I: Take the given mathematics expression.

12 + 45 – 5³ + (18 + 2) * 3 + 10/5 – (13 + 7) + 5⁴ * 4

Step-II: Firstly, calculate the parentheses “()” of the math expression

12 + 45 – 5³ + (20) * 3 + 10/5 – (13 + 7) + 5⁴ * 4

12 + 45 – 5³ + 20 * 3 + 10/5 – 20 + 5⁴ * 4

Step-III: Calculate the exponent terms of the math expression.

12 + 45 – (5 x 5 x 5) + 20 * 3 + 10/5 – 20 + 5⁴ * 4

12 + 45 – 125 + 20 * 3 + 10/5 – 20 + (5 x 5 x 5 x 5) * 4

12 + 45 – 125 + 20 * 3 + 10/5 – 20 + 625 * 4

Step-IV: Now calculate the multiplication and division symbol from left to right.

12 + 45 – 125 + 60 + 10/5 – 20 + 625 * 4

12 + 45 – 125 +60 + 2 – 20 + 625 * 4

12 + 45 – 125 +60 +2 – 20 + 2500

Step-V: Calculate the plus and minus symbols from left to right present in the math expression. 

57 – 125 +60 +2 – 20 + 2500

-68 +60 +2 – 20 + 2500

-8 +2 – 20 + 2500

-6 – 20 + 2500

-26 + 2500

2474

Step-VI: Now write the math expression with the result.

12 + 45 – 5³ + (18 + 2) * 3 + 10/5 – (13 + 7) + 5⁴ * 4 = 2474

To avoid such a large number of steps to solve the math expression, use a PEMDAS calculator.

BODMAS Rule

The BODMAS rule is another way to solve the problems of math expression by solving all the mathematics symbols one by one in a correct sequence. The term BODMAS rule stands for:

• B = Brackets (brackets can be braces or parenthesis)
• O = Order of (exponent “^”)
• D = Division “/”
• M = Multiplication “* or x”,
• A = Addition “+”
• S = Subtraction “-”.

Methods to solve math expression using BODMAS

To solve the problems of the math expression with the help of BODMAS rule, follow the steps below.

1. Firstly, calculate the brackets “(brackets can be braces or parenthesis)” of the math expression even if the parentheses are the last term of the expression.
2. Then calculate the ordered terms (exponent or power) of the math expression if there are more than one exponent terms in the expression solve the leftmost term first.
3. After that, calculate the division and multiplication symbol from left to right present in the math expression.
4. Lastly, calculate the plus and minus symbols from left to right present in the math expression. 

Example: By the BODMAS rule

Evaluate 3 * 4 + 7³ + (12 – 4) / 4 – 8² + 30 * 2 – (3 + 7) – 5 * (2 * 3) + 8 by using the BODMAS rule.

Solution

Step-I: Take the given mathematics expression.

3 * 4 + 7³ + (12 – 4) / 4 – 8² + 30 * 2 – (3 + 7) – 5 * (2 * 3) + 8

Step-II: Firstly, calculate the brackets (brackets can be braces or parenthesis) of the math expression.

3 * 4 + 7³ + (8) / 4 – 8² + 30 * 2 – (3 + 7) – 5 * (2 * 3) + 8

3 * 4 + 7³ + 8 / 4 – 8² + 30 * 2 – (10) – 5 * (2 * 3) + 8

3 * 4 + 7³ + 8 / 4 – 8² + 30 * 2 – 10 – 5 * 6 + 8

Step-III: Calculate the ordered (exponent or power) terms of the math expression.

3 * 4 + (7 x 7 x 7) + 8 / 4 – 8² + 30 * 2 – 10 – 5 * 6 + 8

3 * 4 + 343 + 8 / 4 – 8² + 30 * 2 – 10 – 5 * 6 + 8

3 * 4 + 343 + 8 / 4 – (8 x 8) + 30 * 2 – 10 – 5 * 6 + 8

3 * 4 + 343 + 8 / 4 – 64 + 30 * 2 – 10 – 5 * 6 + 8

Step-IV: Now calculate the division and multiplication symbol from left to right present in the math expression.

12 + 343 + 8 / 4 – 64 + 30 * 2 – 10 – 5 * 6 + 8

12 + 343 + 2 – 64 + 30 * 2 – 10 – 5 * 6 + 8

12 + 343 + 2– 64 + 60 – 10 – 5 * 6 + 8

12 + 343 + 2– 64 + 60 – 10 – 30 + 8

Step V: Solve the addition and subtraction terms from left to right.

355 + 2– 64 + 60 – 10 – 30 + 8

357 – 64 + 60 – 10 – 30 + 8

293 + 60 – 10 – 30 + 8

353 – 10 – 30 + 8

343 – 30 + 8

313 + 8

321

Step-VI: Now write the math expression with the result.

3 * 4 + 7³ + (12 – 4) / 4 – 8² + 30 * 2 – (3 + 7) – 5 * (2 * 3) + 8 = 321

Summary

Now after reading the above post, you can solve any math expression according to the accurate sequence of mathematical symbols. You can grab all the basics of the order of operations by learning this post.