While writing a polynomial expression, the number of parentheses can vary from absolute minimum, like in \(x^3\)\(-3x^2y/4\)\(+(x+y)7\)\(-5y^3\), to extravagant many, like in \((x^3)\)\(-((3(x^2)y)/4)\)\(+((x+y)7)\)\(-(5(y^3)).\) And obviously, the polynomial with a minimum number of parentheses is easier to write as well as read. But when we write a polynomial with lesser number of parentheses, few assumptions get introduced into and we have to agree on these assumptions so that different people do not interpret the same equation in different ways. For example, when we write \((x^3)\)\(-((3(x^2)y)/4)\)\(+((x+y)7)\)\(-(5(y^3))\) with least number of parentheses as \(x^3\)\(-3x^2y/4\)\(+(x+y)7\)\(-5y^3\), someone with deficient understanding may subtract \(5\) from \(7\) to write it as \(x^3\)\(-3x^2y/4\)\(+(x+y)2y^3\) and thereby changing the equation altogether. These underlying assumptions, however, are not too hidden to be taken care of. With proper understanding, most students can manage them on their own. The problem, however, is that students get misguided from an erroneous rule called BODMAS rule or PEMDAS rule. In this article, we will try to understand the error introduced by BODMAS or PEMDAS and then the right way to handle the order of operations in polynomial expressions.

Sometimes or many times (depending on the part of the world you live in) students get formally introduced with something called BODMAS rule to handle polynomial expressions. BODMAS stands for Bracket, Order, Division, Multiplication, Addition, Subtraction. It shows the order in which these operations are to be performed in a polynomial expression. For example, if we need to simplify the polynomial expression \(1\)\(+2\times3\)\(-4/2\)\(+(5-1)\)\(+6^2\), then BODMAS rule tells that we have to first evaluate the expression in the bracket (here, parentheses) so that the given expression becomes \(1\)\(+2\times3\)\(-4/2\)\(+4\)\(+6^2\). Then we need to perform the operation of Order (here, square), so the expression becomes\(1\)\(+2\times3\)\(-4/2\)\(+4\)\(+36\). Next, we need to perform division, so the expression becomes \(1\)\(+2\times3\)\(-2\)\(+4\)\(+36\). Next operation is multiplication, so the expression becomes \(1\)\(+6\)\(-2\)\(+4\)\(+36\). After this, we need to perform Addition, so the expression becomes \(7-42\). At last, we need to perform the operation of subtraction which gives the answer \(-35\).

Another variation of BODMAS is PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) which students sometimes memorize with the help of sentence ‘Please Excuse My Dear Aunt Sally’ or something similar. PEMDAS is more common in the American continent and is equally right or wrong as the BODMAS rule. You may have observed that PEMDAS puts multiplication before division while BODMAS puts division before multiplication. Does it mean that they conflict? The answer is ‘No’ because they both are equally wrong. As we will see below, multiplication and division; addition and subtraction have equal precedence for the order of operation. Unfortunately, students who learn BODMAS or PEMDAS seldom are aware of the equal-precedence of inverse operations.

Let us see an example where a student commits mistake simply because she follows BODMAS rule. One of the simplest examples can be this: Evaluate \(7-2+1\). If the student follows the BODMAS rule, she would perform addition before subtraction and hence her steps would be \(7-2+1\)\(= 7-3\)\(= 4\) while the correct steps are \(7-2+1\)\( = 5+1\)\( = 6\). Interestingly, sometimes a student may compute the expression correctly even after following BODMAS rule. She might proceed with these steps: \(7-2+1\)\( = 7-1\)\(= 6\). This time student is right because she has mentally converted expression \(7-2+1\) into \(7+(-2+1)\). And if she can understand the meaning of the expression like this, she does not need BODMAS rule at all!

Similarly, followers of PEMDAS rule may err with expressions like \(12/3\times2\). They are inclined to perform multiplication before division so may compute it with these steps \(12/3\times2\)\(= 12/6\)\( = 2\) while the correct steps are \(12/3\times2\)\( = 4\times2\)\( = 8\). If you are annoyed with these examples because you always do such calculations right way and never like the incorrect ways shown here, you probably do not use PEMDAS or BODMAS rule. But if you do such mistakes please read on; you need to correct yourself!

So what is the correct order of algebraic operations in polynomial expressions? If we recall why at all we need to have an understanding on the precedence of mathematical operations, we see that it is because we want to write our polynomial expressions with minimum use of parentheses. So when we still see parentheses in an expression, we need to assume that the writer of the expression has kept them because their removal introduces ambiguity. For example, parentheses from \((3+4)\times2\) cannot be removed because then expression becomes \(3+4\times2\) which, owing to other rules of the game, will be evaluated as \(3+4\times2\)\(= 3+8\)\(=11\) while the original meaning was \((3+4)\times2\)\(= 7\times2\)\( = 14\). So when we still see parentheses in expressions, we have to evaluate them first. Another obvious, but commonly ignored, convention of the game is that we have to evaluate expressions from left to right. With this understanding one can now put the correct order of algebraic operations as follows:

- Exponents & Radicals
- Multiplication & Division
- Addition & Subtraction

It is noteworthy that the mathematical operations have tie with corresponding inverse operations. Therefore, multiplication is NOT mandatory to be performed before or after division; instead, the one at left is to be performed at first. This is why \(24/3\times2\) is \(16\) and not \(4\) (which one gets under influence of PEMDAS). Similarly, addition is NOT mandatory to be performed before or after subtraction; instead, the one at left is to be performed at first. So \(9-7+1\) is \(3\) and not \(1\) (which one gets under influence of BODMAS or PEMDAS). Same is the case with exponents and radicals but thankfully BODMAS or PEMDAS do not dictate any order for them.

Hopefully, there is no need of an acronym like BODMAS or anything similar to ‘Please Excuse My Dear Aunt Sally’ to remember above order of operations. But if you do need some help on that, you can consider the fact that more powerful operations are done before less powerful operations. Do you know which of given two operations is more powerful? An exponent is in effect two or more multiplications and a multiplication is two or more additions. Thus, an exponent is more powerful than multiplication and multiplication is more powerful than addition. Similarly, radical can be considered more powerful than division and division more powerful than subtraction.

We can summarize what we have discussed till now as follows:

- You should evaluate what is there within parentheses at first.
- Always evaluate from left to write.
- An operation and its inverse operation have equal precedence and they are evaluated based on which comes first.
- If the need arises to decide, the more powerful operation is to be performed before the less powerful operation.

So far so good but one should not follow above order religiously. Variations are possible and one should use their common sense and their knowledge of algebra to make calculations easier and efficient. Following examples should give you an idea of what it actually means.

**Example 1:** Evaluate \(\sqrt 2 \times \sqrt 2\\\)

**Solution:** If you insist that radical sign has to be evaluated before multiplication, you are not only going to introduce error in your calculation but also going to make your life hell with lengthy calculations. That is because you will first need to approximate \(\sqrt 2\) with a number with as many digits after decimal as you might think fit. This number will only be approximate because actual decimal conversion of \(\sqrt 2\) will never terminate but you will have to stop somewhere. And after multiplying decimal conversions of \(\sqrt 2\), you will reach only close to \(2\) but never exactly equal to \(2\). The correct approach will be to multiply before evaluating radical. Also, you should use formula \(\sqrt a \times \sqrt b = \sqrt {a \times b}\). We, therefore, proceed like this: \(\sqrt 2 \times \sqrt 2\)

**Example 2:** Evaluate \((13^2-11^2)/2\\\)

**Solution:** Do not just jump into parentheses because you think you need to evaluate them first. If you evaluate it as \((13^2-11^2)/2\) \(= (169-121)/2\) \(= 48/2 = 24\), you are not wrong but you could do better. Using formula \(a^2-b^2=(a-b)(a+b)\), you could do this with simpler calculations like this: \((13^2-11^2)/2\) \(=(13-11)(13+11)/2\) \(=2(24)/2=24\)

**Example 3:** Evaluate \(\sqrt {12-3^2}\)\(-(\sqrt 3 – 1)\\\)

**Solution:** No need to perform parentheses before square root. You can evaluate like this: \(\sqrt {12-3^2}\)\(-(\sqrt 3 – 1)\\\) \(= \sqrt {12-9}\)\(-(\sqrt 3 – 1)\\\)\(= \sqrt 3\)\(-(\sqrt 3 – 1)\\\)\( = \sqrt 3-\sqrt 3 + 1\\\)\(= 1 \)

Here we did not care about the order of operation. We did just what was required. And that is the best approach – do whatever is required to simplify the expression following the rules of algebra you already know. No need of BODMAS or PEMDAS!

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