Hierarchy of operations. How to solve combined operations.

Now I’m going to explain to you what is the hierarchy of operations to be able to carry out combined operations with additions, subtractions, multiplications, parentheses and powers at the same time.

We also see how to solve combined operations with parentheses and square brackets, as well as explaining how to solve combined operations with powers and roots.

Combined operations and the hierarchy of operations

When we speak of hierarchy of operations we are talking about the order in which operations must be performed in mathematical expressions where we have more than one operation, additions, subtractions, multiplications, divisions, powers…, that is, in combined operations

In other words, it is the priority that some operations have over others when solving them, taking into account their level within the hierarchy

How combined operations are resolved

When we have expressions where operations are combined, we must start solving the operations at the first level, taking into account the following premises:

• We can’t mix different level operations
• The goal is to reduce the levels to the simplest, which is where there are only additions and subtractions
• Parentheses must be resolved as if they were individual expressions, so the hierarchy of operations must be applied independently of the rest of the expression.

Hierarchy of operations. Priority of mathematical operations

This is the order in which the different operations that may exist in a mathematical expression must be performed:

1. Parenthesis, brackets or keys (resolved from the inside out)
2. Potencies and roots
3. Multiplications and divisions
4. Sum and subtractions

Lower I’ll show you how to apply each one of the levels of operations hierarchy.

Resolved exercises of combined operations

Let’s see an example of how operations are resolved step by step taking into account the hierarchy of operations

Hierarchy of combined operations with additions and subtractions

These operations have no complications, since all operations are at the same level of the hierarchy. All you have to do is operate and you’re done:

The aim is to reduce expressions with operations at various levels down to this level.

Hierarchy of operations combined with additions, subtractions, multiplications and divisions

We are going to incorporate multiplications and divisions:

Now it is necessary to perform first multiplications and divisions, which are at a higher level in the hierarchy:

And we are left with only additions and subtractions, as in the previous case:

One of the most common errors is to solve the equations from left to right without taking into account the hierarchy of operations, that is, mixing operations even if they are not at the same level.

If we start from left to right we would start adding 9+3, which is 12, then multiplying 12.13 which is 156… it would lead us to an incorrect result. Let’s not ever do it

Hierarchy of operations combined with additions, subtractions, multiplications, divisions and powers

We incorporate in this case a power:

In this case, we must solve the power first in order to multiply it by 13:

Once the powers have been removed, we find ourselves in the previous case, so it is resolved in the same way:

In this case we have solved first the power, then the multiplications and divisions and then the additions and subtractions. We are eliminating levels.

Hierarchy of operations combined with additions, subtractions, multiplications, divisions, powers and parentheses

Now we are going to see the case that we have a parenthesis and within the parenthesis we have powers, multiplications and divisions and additions and subtractions:

We have to solve the parenthesis as if it were a separate expression, or in other words, apply the hierarchy of operations within the parenthesis and forget the rest:

We have solved the powers. The next step is to solve the multiplications and divisions within the parenthesis:

Now it only remains to add within the parenthesis:

We have multiplications, additions and subtractions again, so we are like in the second section:

In this case, the first thing we have resolved is the parenthesis, and once resolved, we continue resolving levels of the rest of the expression, taking into account the hierarchy of operations.

Hierarchy of operations combined with additions, subtractions, multiplications, divisions, powers and parentheses with powers

In this case we are going to add another parenthesis with a power:

We first resolve the parenthesis with the power:

And now we resolve the remaining power:

Now we are at the same point as in the previous section, so we resolve the same:

Let’s see once again how we have been following the hierarchy of operations to solve the combined operations.

A trick is not to rush to solve the operations and focus only on the level we want to solve, without modifying the rest of the operation and without skipping steps. In this way, the expression will be simplified.

Combined operations with powers and roots

The powers and roots are at the second level of the hierarchy of operations, above multiplications and divisions and must therefore be resolved before these.

You don’t have to learn at what level each of the operations is, as common sense will tell you what to do, as we’ll see in this example:

In this operation we have additions, subtractions, multiplications and powers.

Let’s forget about power for a moment. We know, from the previous lesson, that before adding and subtracting we have to solve multiplications and divisions.

But in this case, we can’t do the multiplication if we don’t solve the power first. That’s why powers and roots are one level above multiplications and divisions. Do you see why I say it’s common sense?

We therefore resolve the powers first:

We are left with an operation with multiplications, additions and subtractions, so we solve the multiplication:

And finally we make the additions and subtractions:

Exactly the same thing happens with roots. Let’s see it with another example:

We have a root within a division, which cannot be done until the root is resolved. So the first thing to do is to solve the root:

Now it is possible to perform the division:

And finally the additions and subtractions:

I’m not going to stop much longer with powers and roots. As you can see, before doing any multiplication or division, you have to solve the powers and the roots, because they are at a higher level in the hierarchy of operations.

How to solve parenthesis

Let’s now see how to remove parentheses in operations. This time I am referring to parentheses that contain more than one term, since, as you know, there are also parentheses that contain negative numbers, which are placed so as not to have two signs in a row.

Let’s start with a very simple example:

In this case we have a parenthesis with 2 terms. To remove it, we must operate within the parenthesis as if it were an isolated operation. We do the subtraction:

Nos has been left a parenthesis with a term. Therefore, we eliminate it following the rule of signs and we can finish the operation:

When operating within the parenthesis, the hierarchy of operations must also be taken into account. Let’s see it with this other example:

First, we have to solve the inside of the parenthesis, but in this case, we have a multiplication, which we will have to solve the first:

We continue with the additions and subtractions within the parenthesis:

Y to finish, we remove the parenthesis according to the sign in front and finish the operation:

Let’s see now this other example, where we have two parentheses:

Within one of them, we have a multiplication, which we go on to solve, leaving the rest of the operation as it is:

Now, we add and subtract each of the parentheses:

Eliminate the parenthesis according to the sign in front and finish the operation:

We continue with another example, in which the parenthesis is multiplied by a number, that is, it is part of a multiplication:

The first step as always would be to resolve the parenthesis:

And once resolved, we perform the multiplication and then the remaining sums:

In the same way, the parenthesis can be part of a division:

We first resolve the parenthesis:

Y we continue with the division and to finish with the subtraction:

We can also have two parentheses multiplying each other:

In this case, solving each parenthesis separately leaves a simple multiplication:

Vas capturing the procedure? When solving the parentheses in the first place, the operation is simplified and we are left with expressions in which the next step is to solve first multiplications and divisions and finally, additions and subtractions.

Combined operations with parentheses and brackets

We are going to increase the difficulty by one degree and we are going to see now when we have parentheses within other parentheses or better said, parentheses within brackets, since the parentheses that enclose other parentheses are called brackets [].

In the first place, we can have square brackets when we already have a negative number enclosed in parentheses, as in this operation:

If you realize, in this case, resolving the bracket is the same as resolving the parentheses we have been resolving until now. We solve the sum:

Y now we remove the parenthesis taking into account the sign in front of it and finish the operation:

It is something more complicated, to solve brackets, that inside have parentheses with more than one term. In these cases, we have to start by solving the parentheses inside and the brackets will become simple parentheses.

Let’s see it with the following example:

In this operation the brackets are needed because inside we have parentheses with more than one term to solve. Therefore, the first step is to solve the parentheses inside the bracket:

Now we remove parentheses (which are simply removed because they are positive numbers) and the square brackets become parentheses, as they have no parentheses inside.

Nos is now an operation with a parenthesis, which we have to solve, starting with the division it has inside:

We solve the parenthesis:

And finally, we remove the parenthesis and perform the subtraction:

Let’s see another example of how to solve operations combined with parentheses and brackets:

This operation has more than one way to resolve itself. You can try it on your own and then check if your result is the same

I’m going to start by solving the parenthesis inside the brackets:

When resolving the parenthesis (which gives a positive number) the square brackets have become parenthesis.

Now I am going to multiply the second parenthesis:

We solve the parentheses:

Finally we remove parentheses and make the resulting sums:

A last example to finish:

We resolve the parentheses inside the brackets:

We solve the parenthesis with two terms and make the division:

If you realize it’s always the same procedure. To solve the square brackets, first we must solve the parentheses inside them. Then, little by little, we simplify and solve levels of the operations hierarchy.

Possibilities there are infinites. You only have to follow the order established by the hierarchy of operations and simplify step by step.

To learn you need to practice, make mistakes and learn from your mistakes.

So now I encourage you to try to resolve these proposed operations.

Proposed exercises of combined operations with parentheses and powers

Resolve the following operations:

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Now we are going to learn how to solve operations combined with fractions with exercises solved step by step.

We already explained how each of the operations with fractions is performed, when in an expression, we only had to solve one of them at the same time.

What are combined operations with fractions

In this case, with operations combined with fractions, we are going to have to add, subtract, multiply or divide the fractions in the same expression. The different operations are going to be mixed, or rather, they are going to be combined.

To correctly solve this type of exercises, we must perfectly understand how each operation is performed separately: additions and subtractions with the same and different denominator, multiplications and divisions

As with operations with numbers, when we have several operations in the same exercise, we have to follow the rules of the operations hierarchy.

Resolved exercises of combined operations with fractions

We’re going to explain it with a few solved examples of operations combined with fractions. We’ll stop at what you need to know in each of the steps.

Example 1:

We have two operations: addition and multiplication. Well, first we do the multiplication:

We are left with a sum with a different denominator. Now we get a common denominator and we make the sum:

In the end, we simplified the fraction.

Example 2:

In this case we have several operations combined with fractions and also parentheses and brackets.

Where to start? Well, you have to start by removing parentheses and the only way to do it is by starting with the one inside. Remember that when we have several parentheses we have to start from the inside out:

We have already removed the inside parenthesis and we have only one parenthesis left. We proceed to solve it:

Finally, we have one division left, which we solve by multiplying it with a cross:

The result should always be simplified whenever possible. When we explained how to simplify fractions, we saw that 2 methods can be used. When, as in this case, the numbers are relatively high, it is convenient to use the second method, which consists of breaking down the numerator and denominator into factors and then cancelling the factors that are repeated above and below.

View as:

We first break down the numerator:

We keep breaking down the denominator:

Finally we write each number as the product of factors and we cancel those that repeat up and down, leaving the final result:

Example 3:

We are going to increase the difficulty a little more. But don’t worry. You’ll see as if you’re solving step by step, each becomes a little easier:

This time, we can perform more than one operation in the same step, since they do not depend on each other. So we start by multiplying the numerator and dividing the denominator:

Now in the numerator we have 3 fractions left to add and subtract with different denominator. We transform them into a common denominator. In order to do this, we remember that we need to obtain the minimum common multiple of the denominators, in order to obtain their equivalent fractions:

Once all the operations in the numerator and denominator have been performed, all that remains is to divide the final fractions and simplify the result: