Order of Operations: Introduction, Rules, and Examples. In mathematics, order of operations is very important and used widely in order to get the correct result. The order of operations is important because it guarantees that all people can read and calculate a problem in the same way. To avoid the wrong result, we use the order of operations.
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It is the rule that tells the correct sequence of steps for calculating a math expression. In order to remember this order, we use PEMDAS which stands for Parenthesis, Exponent, Multiplication, Division, Addition, and Subtraction. In other words, you must start calculating in any math problem by Parenthesis first, then the exponent, then multiplication and division from left to right, then addition and subtraction from left to right. If there is more than one same operation in a problem solve the leftmost one first, then work right. We can also a complex math problem in which math expression is used by an online PEMDAS Calculator.
PEMDAS is used in the United States, teachers use PEMDAS to remember the order of operations. In Asia, teachers use BODMAS to remember the order of operations. BODMAS stands for Brackets, Order, Division/Multiplication, Addition, and Subtraction.
Order of operations follows some rules. Let us discuss them briefly.
In order of operations, always start with operations contained within parenthesis. Parenthesis is used to group part of an expression. If there is more than one set of parentheses, first solve the leftmost and then right one. Parenthesis are denoted by small brackets ().
Example 1
Solve the parentheses of 4/2 * 3 + (4 + 8) – 23 + (3×6).
Solution
Step 1: solve the leftmost parenthesis first.
4/2 * 3 + (4 + 8) – 23 + (3×6)
4/2 * 3 + (12) – 23 + (3×6)
4/2 * 3 + 12 – 23 + (3×6)
Step 2: Now solve the next parenthesis.
4/2 * 3 + 12 – 23 + (18)
4/2 * 3 + 12 – 23 + 18
Example 2
Solve the parentheses of 7/3 * 3 + (14 – 8) – 3 + (14/2).
Solution
Step 1: solve the leftmost parenthesis first.
7/3 * 3 + (14 – 8) – 3 + (14/2)
7/3 * 3 + (6) – 3 + (14/2)
7/3 * 3 + 6 – 3 + (14/2)
Step 2: Now solve the next parenthesis.
7/3 * 3 + 6 – 3 + (14/2)
7/3 * 3 + 6 – 3 + (7)
7/3 * 3 + 6 – 3 + 7
After parentheses, calculate any exponents present in the expression. Exponents are a way of multiplying a number by itself in power times e.g., 34 is 3 multiplied by itself four times, so you would solve it by multiplying 3*3*3*3. If there is more than one exponent present in that expression solve the leftmost first then the right one. If there is no exponent in an expression ignore E in the PEMDAS and move to the next step.
Example 1
Solve the exponent of 4/23 * 3 + 4 + 8 – 32.
Solution
Step 1: solve the leftmost exponent first.
4/23 * 3 + 4 + 8 – 32
4/(2x2x2) * 3 + 4 + 8 – 32
4/8 * 3 + 4 + 8 – 32
Step 2: Now solve the next exponent.
4/8 * 3 + 4 + 8 – 3×3
4/8 * 3 + 4 + 8 – 9
Example 2
Solve the exponent of 72/3 * 3 + 14 – 82 – 3 + 14/2.
Solution
Step 1: solve the leftmost exponent first.
72/3 * 3 + 14 – 82 – 3 + 14/2
7×7/3 * 3 + 14 – 82 – 3 + 14/2
49/3 * 3 + 14 – 82 – 3 + 14/2
Step 2: Now solve the next exponent.
49/3 * 3 + 14 – 82 – 3 + 14/2
49/3 * 3 + 14 – 8×8 – 3 + 14/2
49/3 * 3 + 14 – 16 – 3 + 14/2
After parenthesis and exponent in order of operations move on and look for any multiplication and division. Remember, the division does not necessarily come before multiplication, these operations are solved from left to right.
Example 1
Solve the multiplication and division of 4/2 * 3 + 4 + 8 – 9/3.
Solution
Step 1: Start from the left and divide the leftmost fraction.
4/2 * 3 + 4 + 8 – 9/3
2 * 3 + 4 + 8 – 9/3
Step 2: Now multiply.
2 * 3 + 4 + 8 – 9/3
6 + 4 + 8 – 9/3
Step 3: Now move to the right and check for any operation related to multiplication or division.
6 + 4 + 8 – 9/3
6 + 4 + 8 – 3
Example 2
Solve the multiplication and division of 27/3 * 3 + 14 – 8 – 3 + 2 x 14/2.
Solution
Step 1: Start from the left and divide the leftmost fraction.
27/3 * 3 + 14 – 8 – 3 + 2 x 14/2
9* 3 + 14 – 8 – 3 + 2 x 14/2
Step 2: Now multiply.
9 * 3 + 14 – 8 – 3 + 2 x 14/2
27 + 14 – 8 – 3 + 2 x 14/2
Step 3: Now move to the right and check for any operation related to multiplication or division.
27 + 14 – 8 – 3 + 2 x 14/2
27 + 14 – 8 – 3 + 28/2
Step 4: Now divide.
27 + 14 – 8 – 3 + 28/2
27 + 14 – 8 – 3 + 14
After calculating parenthesis, exponent, multiplication, and division, our problem becomes very simple to solve as there is only addition and subtraction in the expression. Just like multiplication and division, we will add and subtract from left to right.
Example 1
Solve the addition and subtraction of 6 + 4 + 8 – 3.
Solution
Step 1: Start from the left and add the leftmost term.
6 + 4 + 8 – 3
10 + 8 – 3
Step 2: Now add again.
10 + 8 – 3
18 – 3
Step 3: Now only one term remaining, subtract it.
18 – 3
15
Example 2
Solve the addition and subtraction of 27 + 14 – 8 – 3 + 14.
Solution
Step 1: Start from the left and add the leftmost term.
27 + 14 – 8 – 3 + 14
41 – 8 – 3 + 14
Step 2: Now subtract.
41 – 8 – 3 + 14
33 – 3 + 14
Step 3: Now subtract again
33 – 3 + 14
30 + 14
Step 4: Now only one term remaining, add it.
30 + 14
44
To calculate the order of operations, follow four steps.
Let us take some examples in order to understand how to calculate any math expression according to the order of operation. Order of operations calculator is very essential for the accurate results of such problems.
Example 1
Evaluate 4/2 * 3 + (4 + 8) –32+ (3×6).
Solution
Step 1: Solve the parentheses.
4/2 * 3 + (4 + 8) –32+ (3×6)
4/2 * 3 + (12) –32+ (3×6)
4/2 * 3 + 12–32+ (3×6)
4/2 * 3 + 12 – 32+ (18)
4/2 * 3 + 12 – 32+ 18
Step 2: Solve the exponent.
4/2 * 3 + 12 – 32+ 18
4/2 * 3 + 12 – 3×3+ 18
4/2 * 3 + 12 – 9 + 18
Step 3: Solve the multiplication and division from left to right.
4/2 * 3 + 12 – 9 + 18
2 * 3 + 12 – 9 + 18
6 + 12 – 9 + 18
Step 4: Solve the addition and subtraction from left to right.
6 + 12 – 9 + 18
18 – 9 + 18
9 + 18
27
Example 2
Evaluate 7/14 * 2 + (4 – 8) –62+ (13×2).
Solution
Step 1: Solve the parentheses.
7/14 * 2 + (4 – 8) –62+ (13×2)
7/14 * 2 + (-4) –62+ (13×2)
7/14 * 2 – 4 – 62+ (13×2)
7/14 * 2 – 4 –62+ (26)
7/14 * 2 – 4 –62+ 26
Step 2: Solve the exponent.
7/14 * 2 – 4 –62+ 26
7/14 * 2 – 4 – 6×6 + 26
7/14 * 2 – 4 – 36 + 26
Step 3: Solve the multiplication and division from left to right.
7/14 * 2 – 4 – 36 + 26
0.5 * 2 – 4 – 36 + 26
1 – 4 – 36 + 26
Step 4: Solve the addition and subtraction from left to right.
1 – 4 – 36 + 26
-3 – 36 + 26
-39 + 26
-13
Order of fraction is used to avoid the wrong calculation. It uses PEMDAS (Parenthesis, Exponent, multiplication and Division, Addition and Subtraction) to order the expression. In other words, you must start calculating in any math problem by Parenthesis first, then the exponent, then multiplication and division from left to right, then addition and subtraction from left to right. This topic is not difficult. Once you grab the basic knowledge about this topic you will easily solve any problem related to the order of operations.
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