Order of operations: Defined and Explained with calculations

The order of operation is a frequently used technique in mathematics to solve the expressions with different math symbols and operations. It allows following a correct sequence of operations to calculate the math expressions in the right way.

It is a widely used technique that is used all over the world to get a unique answer to a similar expression. In this post, we will discuss the order of operation with examples and solutions that will help you to understand this topic.

What is the order of operation?

In mathematics, a general technique used to calculate the math’s expressions correctly is said to be the order of operations. It states the correct sequence of the math symbols such as sum, exponent, difference, quotient, product, etc.

There are two well-known orders for solving math expressions.

• PEMDAS
• BODMAS

The first well-known order is said to be the PEMDAS which is greatly used in European countries to calculate the expressions of mathematics correctly. While the BODMAS is generally taught in Asian counties.

(i) PEMDAS Rule

PEMDAS rule is generally used in the European nation. It stands for:

• P stands for Parentheses “()”
• E stands for exponent “^”
• M stands for multiplication “x”
• D stands for division “/”
• A stands for Addition “+”
• S stands for subtraction “-”

Here are a few steps to calculate the math expression accurately with the help of the PEMDAS rule.

1. First of all, take a math expression with different math symbols.
2. Evaluate the parenthesis () available in the math expression. If there is more than one parenthesis (), then solve the leftmost parenthesis () first.
3. After that, evaluate the exponent “^” terms available in the math expression. If there is more than one exponent “^”, then solve the leftmost exponent “^” term first.
4. After evaluating the parenthesis and exponents, evaluate the multiplication and division terms of the math expression from left to right.
5. In the last step, solve the sum and difference operations from left to right.

Below is a solved example of this order to understand this concept accurately.

Example

Evaluate 12 + 15 – 4⁴ – 9³ * 5 + (12 / 4) – 6 (12 * 12) / 6 – 2 by using the PEMDAS rule.

Solution

Step-I: Firstly, write the mathematical expression.

12 + 15 – 4⁴ – 9³ * 5 + (12 / 4) – 6 (12 * 12) / 6 – 2

Step-II: Evaluate the parentheses ().

12 + 15 – 4⁴ – 9³ * 5 + (3) – 6 (12 * 12) / 6 – 2

12 + 15 – 4⁴ – 9³ * 5 + 3 – 6 (144) / 6 – 2

12 + 15 – 4⁴ – 9³ * 5 + 3 – 864 / 6 – 2

Step-III: Now evaluate the exponent terms.

12 + 15 – (4 x 4 x 4 x 4) – 9³ * 5 + 3 – 864 / 6 – 2

12 + 15 – (256) – (9 x 9 x 9) * 5 + 3 – 864 / 6 – 2

12 + 15 – 256 – 729 * 5 + 3 – 864 / 6 – 2

Step IV: Evaluate the product and the quotient terms from left to right.

12 + 15 – 256 – 3645 + 3 – 864 / 6 – 2

12 + 15 – 256 – 3645 + 3 – 144 – 2

Step V: Now evaluate the sum and difference terms from left to right.

27 – 256 – 3645 + 3 – 144 – 2

-229 – 3645 + 3 – 144 – 2

-3874 + 3 – 144 – 2

-3871 – 144 – 2

-4018 – 2

-4020

Step VI: Write the given math expression with the result.

12 + 15 – 4⁴ – 9³ * 5 + (12 / 4) – 6 (12 * 12) / 6 – 2 = -4020

To avoid such a time-consuming calculation, use a PEMDAS calculator. This tool will give the result of any math expression in a couple of seconds with steps.

(ii) BODMAS Rule

The other well-known order for solving math expressions is BODMAS. It stands for:

• B stands for Brackets (brackets can be Parentheses “()”, curly brackets “{}”, or square brackets “[]”)
• O stands for Order (exponent “^”)
• D stands for division “/”
• M stands for multiplication “x”
• A stands for Addition “+”
• S stands for subtraction “-”

Here are a few steps to calculate the math expression accurately with the help of the BODMAS rule.

1. First of all, take a math expression with different math symbols.
2. Evaluate the brackets [], parenthesis (), and braces {} available in the math expression. If there is more than one bracket [], then solve the leftmost bracket [] first.
3. After that, evaluate the order or power “^” terms available in the math expression. If there is more than one order or power “^”, then solve the leftmost order or power “^” term first.
4. After evaluating the brackets and powers, evaluate the division and multiplication terms of the math expression from left to right.
5. In the last step, solve the sum and difference operations from left to right.

Below is a solved example of this order to understand this concept accurately.

Example

Evaluate 10 – 5³ + 14 + 6⁴ / 6 – (11 + 1) + 8 (11 – 12) / 2 + 4 by using the PEMDAS rule.

Solution

Step-I: Firstly, write the mathematical expression.

10 – 5³ + 14 + 6⁴ / 6 – (11 + 1) + 8 (11 – 12) / 2 + 4

Step-II: Evaluate the parentheses ().

10 – 5³ + 14 + 6⁴ / 6 – (12) + 8 (11 – 12) / 2 + 4

10 – 5³ + 14 + 6⁴ / 6 – 12 + 8 (-1) / 2 + 4

10 – 5³ + 14 + 6⁴ / 6 – 12 – 8 / 2 + 4

Step-III: Now evaluate the exponent terms.

10 – (5 x 5 x 5) + 14 + 6⁴ / 6 – 12 – 8 / 2 + 4

10 – 125 + 14 + (6 x 6 x 6 x 6) / 6 – 12 –  8 / 2 + 4

10 – 125 + 14 + 1296 / 6 – 12 –  8 / 2 + 4

Step IV: Evaluate the quotient and the product terms from left to right.

10 – 125 + 14 + 216 – 12 – 8 / 2 + 4

10 – 125 + 14 + 216 – 12 –  4 + 4

Step V: Now evaluate the sum and difference terms from left to right.

-115 + 14 + 216 – 12 –  4 + 4

-101 + 216 – 12 –  4 + 4

115 – 12 –  4 + 4

103 –  4 + 4

99 + 4

103

Step VI: Write the given math expression with the result.

10 – 5³ + 14 + 6⁴ / 6 – (11 + 1) + 8 (11 – 12) / 2 + 4 = 103

Conclusion

The PEMDAS and BODMAS are almost similar in calculations as you have seen in the above post. Now your confusion about solving math expressions correctly must be resolved by this post. Now you can solve any math expression either by BODMAS or PEMDAS rules.