Priority in Performing Operations
- Perform operations within parentheses, brackets, and braces.
- Calculate powers and roots.
- Perform products and quotients.
- Perform addition and subtraction.
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:

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


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


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


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


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


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


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.


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).


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:


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:


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.


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


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:


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


Similarly, we start with the operations within the parentheses:


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


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


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


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


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).


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


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


We start with the additions within the parentheses:


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).



Summarize with AI:








