In mathematics, solving expressions in the correct manner relies on adhering to a specific order called the order of operations. The rule of PEMDAS offers a systematic framework to obtain correct solutions in simplifying complicated expressions by specifying what operations to conduct first. PEMDAS is an acronym for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction, and abiding by this sequence of operations ensures avoiding errors that might result from carrying out operations in the wrong manner. Knowing PEMDAS is particularly important for students aiming to take U.S. competitive exams such as the SAT, ACT, and GRE, where understanding mathematical principles, such as order of operations, is fundamental in scoring high. This article looks at the importance of the PEMDAS rule, illustrates its usage through step-by-step examples, contrasts it with the BODMAS rule, and offers valuable tips to maximize problem-solving efficiency.

PEMDAS

The correct order is not followed it is likely to get numerous or distinct answers.

The pemdas full form is as follows:

• P stands for Parentheses in the expression [{()}].
• E stands for Exponents in the expression (\(x^{2}\), \(x^{3}\), etc).
• MD stands for Multiplication and Division(× and ÷) starting from left to right.
• AS denotes Addition and Subtraction (+ and -) starting from left to right.

PEMDAS Rule

In the previous header, we learn about the pemdas meaning with its full form and its need to use. Moving ahead let us understand the rule in terms of the order in detail. In the term PEMDAS, each letter represents an operation in mathematics and the order in which letters are arranged directs us to the order to be followed while solving different parts of the given expression.
The rules state that:

• P-Parentheses; Start simplifying the expression with the terms in parentheses. That is to simplify any type of parentheses or brackets like; ( ),{ } or [ ].
• E-Exponents; Once the parentheses are simplified, the exponential or the power term should be computed as the next step. For example, solve the terms like; \(a^{2}\), \(\sqrt{9}\), \(4^{2}\), and so on.
• MD-Multiplication OR Division; Once the parentheses and exponents are solved. The next step to solving is to pick for multiplication or division(× OR ÷), whichever comes first starting from the left to right.
• AS- Addition OR Subtraction; At last complete the simplification by performing the addition or subtraction(+ OR -) while driving from left to right whichever comes first.

Note: In the PEMDAS Rule, multiplication OR division, addition OR subtraction have equal priority i.e we can do any of the two first. To memorise this abbreviation, the phrase used is “Please Excuse My Dear Aunt Sally”.

Steps to Apply PEMDAS Rule

With the complete outline of the full form of PEMDAS in maths with its rule pattern, it’s time to understand the step to solve any expression using the rule.

Simplify the expression below using the pemdas rule: \(4\times4\div\left(6-4\right)-2^2+4\times5\)

Step 1: Always remember to simplify the terms inside the parentheses symbols. That is to start with simplifying brackets and braces like ( ), { }.

\(4\times4\div2-2^2+4\times5\)

Here there was only one bracket, in the case of nested grouping symbols, start simplifying from inside to out.

Step 2: Next, solve the exponential terms first before executing any of the 4(multiplication, division, addition and subtraction) fundamental arithmetic operations like solving Exponential Functions.

\(4\times4\div2-4+4\times5\)

Step 3: As multiplication and division hold a higher level of priority than addition and subtraction. After the exponential terms are resolved using exponent rules, pick the terms related to multiplication and/or division whichever arrives first from left to right.

\(16\div2-4+4\times5\)

Applying the step one more time as the multiplication and division operator is still present.

\(8-4+20\)

Step 4: Wind up the calculation by the addition and/or subtraction of the term whichever arrives first from left to right.

=8-4+20

=-4+20

=16

That is; \(4\times4\div\left(6-4\right)-2^2+4\times5=16\).

In case any of these steps are missed or shuffled, i..e if the multiplication is performed before the exponents, the final result will be different.

Mistakes to avoid in PEMDAS Rule

At this point of the discussion, we can clearly state that the PEMDAS formula is nothing but the directive rule of calculations using which we solve difficult equations in a step-by-step manner. For any rule, it is important to understand the do and don’t. When to apply, when not to apply and so on. Let us look at this aspect of the rule as well.

• The rule is applied when the equation or the expression includes more than one operation.
• There is a set of rules that need to be followed when operating the PEMDAS method. If any of the steps are missed or revised, this will lead to a miscalculation of the outcome.
• While solving a particular equation the presence of multiple brackets mostly leads to confusion. In such a case the right step is to start with the innermost bracket followed by the outer ones.

Difference between PEMDAS and BODMAS

Similar to the pemdas rule in algebra there are rules like GEMS (Grouping, Exponents, Multiply/Divide, Add/Subtract), BODMAS (Brackets, Orders, Divide, Multiply, Add, Subtract) preferably used in the UK, BEDMAS (Brackets, Exponents, Divide, Multiply, Add, Subtract) commonly preferred in Canada. All these acronyms are the same only. Let us understand the difference between bodmas and pemdas with a tabular comparison.

Solved Examples of PEMDAS

Now we know when and why we use the pemdas rule, the full form along with the rules. It’s time to practise some solved examples, as the rule can be best learnt with the practice of examples only. Although the solutions are given, give it a try and very you answer with the solution.

Solved Example 1: Simplify the expression using the PEMDAS rule.

16÷(2+2)−8×10=?

Solution: Given; 16÷(2+2)−8×10=?

Using the rule:

Solve the brackets first, followed by the exponent terms if present and then proceed with multiplication/division followed by addition or subtraction.

16÷(2+2)−8×10

=16÷4−8×10

=4−8×10

=4−80

16÷(2+2)−8×10=-76

Solved Example 2: Estimate the out of the equation;16+10×2.5−18÷9 by applying the PEMDAS formula.

Solution: Given, 16+10×2.5−18÷9=?

In the absence of any parentheses and exponents, directly start with the multiplication and division.

16+10×2.5−18÷9

Here, 10×2.5=25 and 18÷9=2

=16+25−2

Now add and subtract the term.

=39

That is, 16+10×2.5−18÷9=29

Solved Example 3: Simplify the expression using the PEMDAS expansion.

\(\left(2+3\right)^2−4\left(8−6\right)\)

Solution: Given that,\(\left(2+3\right)^2−4\left(8−6\right)\)=?

On solving the parentheses, we get;

\(\left(5\right)^2−4\left(2\right)\)

Next, consider the exponent term;

\(25−4\times2\)

Now perform the multiplication;

25−8

Lastly carry out the subtraction;

=17

\(\left(2+3\right)^2−4\left(8−6\right)=17\)

Solved Example 4: Simplify the expression applying the PEMDAS formula. \(−20+22−32\div\left(64\div2^3\right)\)

Solution: Given that,\(−20+22−32\div\left(64\div2^3\right)\)=?

The order of operations is as follows; parentheses, exponents, multiplication, division, addition, and subtraction.

Here the parentheses term is \(\left(64\div2^3\right)\)

We need to follow the rule within the bracket as well. That is to solve the exponent first followed by division.

\(\left(64\div8\right)=8\)

The expression left with us is;

\(−20+22−32\div8\)

Perform the division operation;

−20+22−4

Lastly add and subtract the numbers to obtain the answer;

=-2

That is \(−20+22−32\div\left(64\div2^3\right)=-2\).

Solved Example 5: Simplify the expression using the PEMDAS rule.

\(\sqrt{2+7}-10+3\).

Solution: Given that, \(\sqrt{2+7}-10+3=?\)

As per rules, first, we will simplify the root.

\(\sqrt{2+7}=\sqrt{9}=3\)

The expression now is, 3-10+3

Perform the subtraction and addition operation for the final answer.

=-4

\(\sqrt{2+7}-10+3=-4\)

In summary, it is important for students sitting for U.S. competitive exams like the SAT, ACT, GRE, and GMAT to know and use the PEMDAS rule since mathematical accuracy can make a big difference in test scores. By learning the proper order of operations — Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction — students can effectively solve intricate problems and prevent frequent mistakes due to performing operations in the wrong order. Establishing a strong PEMDAS foundation not only increases problem accuracy but also cultivates confidence when solving difficult quantitative passages on entrance exams. Seeking admission to prestigious colleges or competitive advantage in master's exams, having a well-developed PEMDAS could prove to make all the difference in scoring well.