PAPOMUDAS: HOW TO SOLVE THEM AND EXERCISES - MATHS - 2025

The papomudas is a method of solving algebraic expressions. Its acronyms indicate the order of priority of the operations: parentheses, powers, multiplication, division, addition and subtraction. Using this word you can easily remember the order in which an expression composed of several operations must be solved.

Generally, in numerical expressions you can find several arithmetic operations together, such as addition, subtraction, multiplication and division, which can also be fractions, powers and roots. To solve them it is necessary to follow a procedure that guarantees that the results will be correct.

An arithmetic expression that is made up of a combination of these operations must be solved according to priority order, also known as the hierarchy of operations, established long ago in universal conventions. Thus, all people can follow the same procedure and obtain the same result.

characteristics

Papomudas is a standard procedure that establishes the order to be followed when solving an expression, which is composed of a combination of operations such as addition, subtraction, multiplication and division.

This procedure establishes the order of priority of an operation in relation to the others at the time they will be results; that is, each operation has a shift or hierarchical level to be resolved.

The order in which the different operations of an expression must be solved is given by each acronym of the word papomudas. Thus, you have to:

1- Pa: parentheses, brackets or braces.

2- Po: powers and roots.

3- Mu: multiplications.

4- D: divisions.

5- A: additions or additions.

6- S: subtractions or subtractions.

This procedure is also called in English as PEMDAS; To easily remember this word, it is associated with the phrase: "Please Excuse My Dear Aunt Sally", where each initial letter corresponds to an arithmetic operation, in the same way as papomudas.

How to solve them?

Based on the hierarchy established by papomudas to solve the operations of an expression, it is necessary to fulfill the following order:

- First, all operations that are within grouping symbols, such as parentheses, braces, square brackets, and fraction bars, must be solved. When there are grouping symbols within others, you must start calculating from the inside out.

These symbols are used to change the order in which operations are solved, because what is inside them must always be solved first.

- Then the powers and roots are solved.

- In third place the multiplications and divisions are solved. These have the same order of priority; therefore, when these two operations are found in an expression, the one that appears first must be solved, reading the expression from left to right.

- In the last place the additions and subtractions are solved, which also have the same order of priority and, therefore, the one that appears first in the expression is solved, read from left to right.

- Operations should never be mixed when read from left to right, the order of priority or hierarchy established by the papomudas should always be followed.

It is important to remember that the result of each operation must be placed in the same order in relation to the others, and all intermediate steps must be separated by a sign until the final result is reached.

Application

The papomudas procedure is used when you have a combination of different operations. Taking into account how they are solved, this can be applied in:

Expressions containing addition and subtraction

It is one of the simplest operations, because both have the same order of priority, in such a way that it must be solved starting from the left to the right in the expression; for example:

22 -15 + 8 +6 = 21.

Expressions containing addition, subtraction, and multiplication

In this case, the operation with the highest priority is multiplication, then the additions and subtractions are solved (the one that is first in the expression). For example:

6 _* 4 - 10 + 8 _* 6 - 16 + 10 _* 6

= 24 -10 + 48 - 16 + 60

= 106.

Expressions containing addition, subtraction, multiplication, and division

In this case you have a combination of all the operations. You start by solving the multiplication and division that have higher priority, then the addition and subtraction. Reading the expression from left to right, it is solved according to its hierarchy and position within the expression; for example:

7 + 10 _* 13 - 8 + 40 ÷ 2

= 7 + 130 - 8 + 20

= 149.

Expressions containing addition, subtraction, multiplication, division, and powers

In this case, one of the numbers is raised to a power, which within the priority level must be solved first, to then solve the multiplications and divisions, and finally the additions and subtractions:

4 + 4 ² _* 12 - 5 + 90 ÷ 3

= 4 + 16 _* 12 - 5 + 90 ÷ 3

= 4 + 192 - 5 + 30

= 221.

Like the powers, the roots also have the second order of priority; Therefore, in expressions that contain them, multiplication, division, addition and subtraction must be solved first:

5 _* 8 + 20 ÷ √16

= 5 _* 8 + 20 ÷ 4

= 40 + 5

= 45.

Expressions that use grouping symbols

When signs such as parentheses, braces, square brackets and fraction bars are used, what is inside these is resolved first, regardless of the order of priority of the operations that it contains in relation to those outside it, as if It will be a separate expression:

14 ÷ 2 - (8 - 5)

= 14 ÷ 2 - 3

= 7 - 3

= 4.

If there are several operations within it, they must be resolved in hierarchical order. Then the other operations that make up the expression are solved; for example:

2 + 9 * (5 + 2 ³ - 24 ÷ 6) - 1

= 2 + 9 * (5 + 8 - 4) - 1

= 2 + 9 * 9 - 1

= 2 + 81 - 1

= 82.

Some expressions use grouping symbols within others, such as when the sign of an operation needs to be changed. In these cases, you must begin by solving from the inside out; that is, by simplifying grouping symbols that are in the center of an expression.

Generally, the order to solve operations contained within these symbols is: first solve what is inside parentheses (), then brackets and last the braces {}.

90 - 3 _*

= 90 - 3 _*

= 90 - 3 _* 24

= 90 - 72

= 18.

Exercises

First exercise

Find the value of the following expression:

20 ² + √225 - 155 + 130.

Solution

Applying the papomudas, the powers and roots have to be solved first, and then addition and subtraction. In this case, the first two operations belong to the same order, so the one that is first is solved, starting from left to right:

20 ² + √225 - 155 + 130

= 400 + 15 -155 + 130.

Then you add and subtract, starting from the left too:

400 + 15 -155 + 130

= 390.

Second exercise

Find the value of the following expression:

Solution

It begins by solving the operations that are inside the parentheses, following the hierarchical order that these have according to the papomudas.

The powers of the first parentheses are solved first, then the operations of the second parentheses are solved. Since they belong to the same order, the first operation of the expression is solved:

=

=

=.

As the operations within the parentheses have already been resolved, now we continue with the division that has a higher hierarchy than the subtraction:

=.

Finally, the parenthesis that separates the minus sign (-) from the result, which in this case is negative, indicates that these signs must be multiplied. Thus, the result of the expression is:

= 171.

Third exercise

Find the value of the following expression:

Solution

You start by solving the fractions that are inside the parentheses:

Inside the parentheses there are several operations. The multiplications are solved first and then the subtractions; In this case, the fraction bar is considered as a grouping symbol and not as a division, so the operations of the upper and lower part must be solved:

In hierarchical order, the multiplication must be solved:

Finally, the subtraction is solved:

References

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2. Aponte, G. (1998). Fundamentals Of Basic Mathematics. Pearson Education.
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5. Huffstetler, K. (2016). The Story of the Order of Operations: Pemdas. Create Space Independent.
6. Madore, B. (2009). GRE Math Workbook. Barron's Educational Series,.
7. Molina, FA (sf). Azarquiel Project, Mathematics: First cycle. Azarquiel Group.