Priority in Performing Operations

  • Perform operations within parentheses, brackets, and braces.
  • Calculate powers and roots.
  • Perform products and quotients.
  • Perform addition and subtraction.

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Types of Combined Operations

We can classify operations in the following way, which will help us better understand the order of operations:

Combined Operations Without Parentheses

1.1 Combination of Addition and Subtraction

Consider, for example, the following combined operation:

To perform it, we start from the left and carry out operations as they appear:

1.2 Combination of Addition, Subtraction, and Multiplication

Now consider the following operation, which includes multiplication:

We perform the products first since they have higher priority.

Then we perform addition and subtraction (from left to right).

1.3 Combination of Addition, Subtraction, Multiplication, and Division

Consider an operation that also includes division:

We perform products and quotients in the order we encounter them, from left to right, because both operations have the same priority.

Finally, we perform addition and subtraction, also from left to right.

1.4 Combination of Addition, Subtraction, Multiplication, Division, and Powers

Now let's look at the following operation that includes powers:

First we perform the powers since they have the highest priority.

Then we perform products and quotients.

Finally, we perform addition and subtraction from left to right.

Combined Operations With Parentheses

Now let's consider operations that use parentheses. For example:

We first perform the operations contained in the parentheses. We start with powers and then multiplications and divisions:

We continue with addition and subtraction within the parentheses. Note that we can remove the parentheses once we have performed all operations inside them:

Combined Operations With Parentheses and Brackets

Finally, we'll look at operations where we combine everything: parentheses, brackets, powers, multiplication, division, addition, and subtraction.

First we operate with powers, products, and quotients within the parentheses.

We perform addition and subtraction within the parentheses.

We operate within the brackets.

We multiply.

We subtract and add.

Proposed Exercises

Perform the following operations:

1

Solution

First we perform the multiplications, then we perform addition and subtraction:

 

2

Solution

First we perform the divisions, then we perform the addition:

 

3

Solution

First we perform the multiplications, then we perform addition and subtraction:

 

4

Solution

Now we perform the division first. Then we continue with addition and subtraction:

 

5

Solution

Now we perform the multiplication first. Then we continue with addition and subtraction:

 

6

Solution

Now we perform the divisions first. Then we continue with subtraction:

 

7

Solution

Since we now have parentheses, we first perform the operations within them. Remember to follow the same order of operations within each parenthesis (that is, we perform the multiplication first). Also, remember that two parentheses together, without a sign between them, is a way to indicate multiplication.

 

8

Solution

Since we now have parentheses, we first perform the operations within them. Remember to follow the same order of operations within each parenthesis (that is, we perform the multiplication first).

 

9

Solution

Just as in the previous exercise, we first perform the operations within the parenthesis. Once we finish, we execute the rest of the operations in their normal order:

 

10

Solution

Just as in the previous exercise, we first perform the operations within the parenthesis as well as the multiplications. Once we finish, we execute the rest of the operations in their normal order:

 

11

Solution

In the same way, we must first perform the operations within the parentheses. Then we perform the powers, followed by multiplication, and finish with addition.

 

12

Solution

Here we must perform the operations of the parentheses (since they are inside the brackets). Then we follow the usual order:

 

13

Solution

Similarly to the previous exercise, we first perform the operations within the parentheses (since they are more "interior"). Then we continue with the brackets, since they are inside the braces:

 

14

Solution

This exercise is very similar to the previous ones. We first perform the operations within the parentheses. Also observe that .

 

15

Solution

Similarly, we start with the operations within the parentheses:

 

16

Solution

We start with the division within the brackets. Remember that division of a positive number by a negative number results in a negative number.

 

17

Solution

Here we must remember that an odd power of negative numbers is a negative number. Therefore:

 

18

Solution

This exercise is very similar to the previous ones. We start with the operations within the parentheses:

 

19

Solution

We proceed as in the previous exercises, starting with the operations within the parentheses:

 

20

Solution

In this case, we first add fractions within the parentheses. Then we perform fraction subtraction. It's important that we don't work with fractions as divisions (although we can, the only drawback is that we would work with decimals).

 

21

Solution

This is similar to the previous exercise. First we perform fraction addition within the parenthesis, then we perform fraction division.

 

22

Solution

Just as in the previous exercise, we first perform the subtractions within the parentheses.

 

23

Solution

We start with the additions within the parentheses:


24

Solution

Similarly to the previous exercises, we follow the order we established at the beginning (starting with operations within parentheses and finishing with addition and subtraction).

 

25

Solution

Summarize with AI:

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Agostina Babbo

Agostina Babbo is an English and Italian to Spanish translator and writer, specializing in product localization, legal content for tech, and team sports—particularly handball and e-sports. With a degree in Public Translation from the University of Buenos Aires and a Master's in Translation and New Technologies from ISTRAD/Universidad de Madrid, she brings both linguistic expertise and technical insight to her work.