por What is the papomuda?
He papomudas is a procedure for solving algebraic expressions. Its initials 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 made up 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 of 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 the priority of 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
The papomudas is a standard procedure that establishes the order that must be followed when solving an expression, which is made up of a combination of operations such as addition, subtraction, multiplication and division.
With this procedure the order of priority of an operation is established in relation to the others at the moment in which they will be results; that is, each operation has a turn or hierarchical level to be resolved.
The order in which the different operations of an expression must be resolved is given by each acronym of the word papomudas. In this way, you have to:
Pa: parentheses, brackets or braces.
Po: powers and roots.
Mu: multiplications.
D: divisions.
A: additions or sums.
S: subtractions or subtractions.
This procedure is also called in English as PEMDAS; to easily remember this word 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 the papomudas.
How to solve them?
Based on the hierarchy established by the papomudas to resolve the operations of an expression, it is necessary to comply with the following order:
All operations inside grouping symbols, such as parentheses, braces, square brackets, and fraction slashes, must first be solved. When grouping symbols exist inside others, you must start calculating from the inside out.
These symbols are used to change the order in which the operations are solved, because what is inside them must always be solved first.
Then the powers and roots are solved.
Thirdly, multiplications and divisions are solved. These have the same order of priority; For this reason, 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, read from left to right, is solved. The operations should never be mixed when being 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 resolved, this can be applied to:
Expressions that contain addition and subtraction
It is one of the simplest operations, because both have the same order of precedence, 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 that contain addition, subtraction and multiplication
In this case, the operation with the highest priority is multiplication, then the addition and subtraction are solved (whichever 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 we have a combination of all operations. Begin by solving multiplication and division, which have top priority, then addition and subtraction. Reading the expression from left to right, it is resolved 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 + 42 * 12 – 5 + 90 ÷ 3
= 4 + 16 * 12 – 5 + 90 ÷ 3
= 4 + 192 – 5 + 30
= 221.
Like powers, roots also have the second order of precedence; for this reason, in expressions that contain them, they must be solved first than multiplications, divisions, additions and subtractions:
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 slashes are used, what is inside them is resolved first, regardless of the order of priority of the operations that it contains in relation to those that are outside it, as if they were It would 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 + 23 – 24 ÷ 6) – 1
= 2 + 9 * (5 + 8 – 4) – 1
= 2 + 9 * 9 – 1
= 2 + 81 – 1
= 82.
In some expressions, grouping symbols are used inside others, such as when it is necessary to change the sign of an operation. In those cases, you should start solving from the inside out; that is, by simplifying the 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 finally the braces {}.
90 – 3*[12 + (5*4) – (4*2)]
= 90 – 3* [12 + 20 – 8]
= 90 – 3 * 24
=90 – 72
= 18.
Exercises
First exercise
Find the value of the following expression:
202 + √225 – 155 + 130.
Solution
Applying the papomudas, powers and roots must be solved first, and then addition and subtraction. In this case, the first two operations belong to the same order, so the one that comes first is solved, starting from left to right:
202 + √225 – 155 + 130
= 400 + 15 -155 + 130.
Then you add and subtract, starting from the left as well:
400 + 15 -155 + 130
= 390.
second exercise
Find the value of the following expression:
[- (63 – 36) ÷ (8 * 6 ÷16)].
Solution
It begins by solving the operations that are inside the parentheses, following the hierarchical order that they have according to the papomudas.
First the powers in the first parenthesis are resolved, then the operations in the second parentheses are resolved. Since they belong to the same order, the first operation of the expression is solved:
[- (63 – 36) ÷ (8 * 6 ÷16)]
= [- (216 – 729) ÷ (8 * 6 ÷16)]
= [- (216 – 729) ÷ (48 ÷16)]
= [- (-513) ÷ (3)].
Since the operations inside the parentheses have already been solved, now continue with the division that has a higher hierarchy than the subtraction:
[- (-513) ÷ (3)] = [- (-171)].
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)] = 171.
third exercise
Find the value of the following expression:
Solution
You begin by solving the fractions that are inside the parentheses:
Inside the parentheses 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:
By hierarchical order, the multiplication must be solved:
Finally, the subtraction is solved: