Order Of Operations – Definition, Steps, FAQs, Examples

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    It’s not uncommon for learners to face various types of mathematical equations. For instance, when solving a mathematical equation like 52 x (2 + 3) – 10/2 + 1, learners could go straight and do the math from left to right and get their final answer as 22.5.

    But do you know that the correct answer is 121? If you do, then you probably understand the rules of the order of operations. However, if you’re unsure how we got that answer, keep reading this article to know the order of operations definition and how to solve questions using this rule.

    What Is the Order of Operations in Math?

    The order of operations in math refers to the standard practice everyone should follow when calculating math equations that involve different operations like +, -, x, and ÷. Therefore, when solving an equation including many operations, it is essential to follow the rule of the order of operations. The rule simply explains that every calculation should follow the stipulated order of operations: PEMDAS. This acronym stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction, and you should memorize the sequence.

    Following the mathematical order of operations as recommended is essential when evaluating a mathematical equation. Actually, the order of operations math is a collection of rules that specifies the recommended order for carrying out specific arithmetic procedures. These principles should be used by every student to guarantee proper and consistent expression evaluation and solving.

    Steps in Computing the Math Order of Operations

    A standard rule of thumb is to remember to read the problem from left to right when doing any multiplication or division. Similarly, add or subtract numbers in the problem by reading them from left to right.

    You must do everything in the proper sequence to get the correct answer in a mathematical equation. Use parentheses to define the order in which you should carry out operations if you are unclear about the order.

    Parentheses

    When applying the order of operations, any calculations should be done within the parenthesis [()] first. For instance, when solving 52 x (2 + 3) – 10/2 + 1, the first action is to deal with the numbers in brackets (2+3) = 5. So, the equation becomes 52 x 5 – 10/2  + 1.

    Exponents

    After handling parentheses, there is a need to handle exponents. Exponents refer to all superscripts in an equation. A number is multiplied by itself several times when it has an exponent. For instance, 52 means 5 x 5 = 25, which is 5 in two places. So, the equation becomes 25 x 5 – 10/2 + 1.

    Multiplication and Division

    It is necessary to do multiplication and division first in the standard order of operations. In this case, 25 x 5 = 125, and 10/2 = 5. Therefore, the new equation becomes 125 – 5 + 1.

    Addition and Subtraction

    The final step is to add/subtract values. In this case, 125 – 5 = 120, and 120 + 1 = 121. Therefore, the answer to 52 x (2 + 3) – 10/2 + 1 = 121.

    Order of Operations Rules

    When solving an equation including numerous operations, you must follow specific rules. The rules are usually named in a short-form format as PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction.

    • P = Parenthesis. You should do any computations within the parenthesis first.
    • E = Exponents. Analyze all exponents next. Exponents, which means raising the power, could look like 2, 3, 4, 5, 6, and more.
    • M & D = Multiplication and Division. Perform any multiplication or division in the equation from left to right.
    • A & S = Carry out any addition or subtraction in the equation from the left side to the right side.

    Order of Operations – PEMDAS vs. BODMAS

    PEMDAS and BODMAS are two acronyms that are valuable in helping learners understand the concept of order of operation better. While the British use BODMAS, Americans use PEMDAS. BODMAS is an acronym for Bracket, Open, Division, Multiplication, Addition, and Subtraction. On the other hand, PEMDAS stands for Parenthesis, Exponent, Multiplication, Division, Addition, and Subtraction.

    Mathematicians may use both acronyms to help learners quickly recall the proper order of operations to solve a specific mathematical problem. The first two steps of PEMDAS and BODMAS are performed in a slightly different sequence. But you will get the same outcome even if you use an online order of operations calculator. Although they both ultimately solve an identical set of equations; when calculating using either PEMDAS or BODMAS, you should always acknowledge the sequence of operations to get the right results.

    How to Use the Order of Operations?

    When solving a mathematical problem, there’s a particular sequence in which you must carry out operations. The sequence is to:

    1. Solve all math problems inside the brackets.
    2. Simplify the exponents. 
    3. Do all necessary multiplication and division problems.
    4. Solve addition and subtraction problems from left to right.

    Some Ways to Remember the Order of Operations

    Here comes the fun part in the order of math operations. The abbreviation PEMDAS (parentheses, exponents, multiplication, division, addition, subtraction) is used to assist the US pupils recall this sequence of operations. Meanwhile, educators use the BODMAS, which stands for “brackets, orders, division, multiplication, and subtraction.”

    You can use a mnemonic to help you recall the sequence of operations, such as “Please Excuse My Dear Aunt Sally.” Also, constant practice will always top the list of ways to remember the order of operations. The repetitions will cement the steps of order of operations in a learner’s memory.

    Solved Examples on the Order of Operations

    It’s not enough to give ony rhetoric on how to fix the algebraic order of operations. So, here are three solved examples using the order of operations:

    Example 1

    • Solve 1² x (2+ 3) – 6/2 + 5

    Parenthesis = 1² x (2+ 3) – 6/2 + 5 = 1² x (5) – 6/2 + 5

    Exponent = 1² x (5) – 6/2 + 5 = 1 x (5) – 6/2 + 5

    Multiplication = 1 x (5) – 6/2 + 5 = 5 – 6/2 + 5

    Division = 5 – 6/2 + 5 = 5 – 3 + 5

    Addition and Subtraction = 5 – 3 + 5 = 7

    Example 2

    • Solve 2³ – 5 (3+1) + 1 – 3(4 x 2)

    Parenthesis = 2³ – 5 (3+1) + 1 – 3(4 x 2) = 2³ – 5 x 4 + 1 – 3 x 8

    Exponent = 2³ – 5 x 4 + 1 – 3 x 8 = 8 – 5 x 4 + 1 – 3 x 8

    Multiplication = 8 – 5 x 4 + 1 – 3 x 8 = 8 – 20 + 1 – 24

    Addition and Subtraction = 8 – 20 + 1 – 24 = – 35

    Example 3

    • Solve for 4² + 8 (3-2) – 8 + 8 ÷ 2

    Parenthesis = 4² + 8 (3-2) – 8 + 8 ÷ 2 = 4² + 8 x 1 – 8 + 8 ÷ 2 

    Exponent = 4² + 8 x 1- 8 + 8 ÷ 2 = 16 + 8 x 1 – 8 + 8 ÷ 2

    Multiplication = 16 + 8 – 8 + 8 ÷ 2

    Division = 16 + 8 – 8 + 8 ÷ 2 = 16 + 8 – 8 + 4

    Addition and Subtraction = 16 + 8 – 8 + 4 = 20

    Frequently Asked Questions On Order Of Operations

    What is the order of operations in math?

    The orders of operation in math refer to a collection of rules known as PEMDAS — Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction. It shows you the right way to go about the whole computing process. In a simple breakdown, the order of operations goes in the following format:

    1. Calculate the operation in the brackets.
    2. Check the exponents.
    3. When multiplying or dividing, always work from left to right.
    4. When adding or subtracting, always work from left to right.

    Are PEMDAS and BODMAS the same?

    Yes, PEMDAS and BODMAS are the same. PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. On the other hand, BODMAS stands for Bracket, Open, Division, Multiplication, Addition, and Subtraction. Both are used in order of operations, and there isn’t much difference.

    The only difference is that while PEMDAS is used in the USA, BODMAS is used in Britain. Also, for BODMAS, division comes before multiplication, unlike with PEMDAS where multiplication comes before division. At the end of the day, both acronyms produce the same outcome.

    How to Apply the Order of Operations with Integers?

    Computing mathematical problems with the order of operations is not as challenging as it looks. All you need to do is know what PEMDAS stands for. You can check the examples section of this article to know more about how to make such calculations.

    How to Solve the Order of Operations?

    The order of operations can be solved using PEMDAS or BODMAS. First, solve the problems in the parenthesis or bracket, then solve the exponents, handle multiplications before divisions, and finally solve additions before subtractions.

    How to Apply the Order of Operations with Exponents?

    When doing the order of operations with exponents, always acknowledge that an exponent means one number multiplied by the number of times in the exponent. For instance, 23, which is 2 with an exponent of 3, means 2 x 2 x 2 = 8. So, 23 + 5 = 2 x 2 x 2 + 5 = 8 + 5 = 13.

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