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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.

Índice de Contenidos

  • 1 Combined operations and the hierarchy of operations
  • 2 How combined operations are resolved
  • 3 Hierarchy of operations. Priority of mathematical operations
  • 4 Resolved exercises of combined operations
  • 5 Hierarchy of combined operations with additions and subtractions
  • 6 Hierarchy of operations combined with additions, subtractions, multiplications and divisions
  • 7 Hierarchy of operations combined with additions, subtractions, multiplications, divisions and powers
  • 8 Hierarchy of operations combined with additions, subtractions, multiplications, divisions, powers and parentheses
  • 9 Hierarchy of operations combined with additions, subtractions, multiplications, divisions, powers and parentheses with powers
  • 10 Combined operations with powers and roots
  • 11 How to solve parenthesis
  • 12 Combined operations with parentheses and brackets
  • 13 Proposed exercises of combined operations with parentheses and powers
  • 14 What are combined operations with fractions
  • 15 Resolved exercises of combined operations with fractions
    • 15.1 Example 1:
    • 15.2 Example 2:
    • 15.3 Example 3:

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:

jerarquía de operaciones

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:

orden de operaciones matemáticas

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

jerarquía de operaciones matemáticas

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

jerarquía de operaciones potencias y raíces

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:

hierarchía de operaciones

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

Which is the hierarchy of combined operations

division or multiplication priority

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

la jerarquía de operaciones

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:

hierarchización de operaciones

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:

operaciones con paréntesis agrchetes y llaves

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

orden de operaciones definición

Now it only remains to add within the parenthesis:

que se hace primero la suma o la multiplicación

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

orden de operaciones ejemplos

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:

operaciones combinadas con potencias

We first resolve the parenthesis with the power:

ley de la jerarquía matemáticas

And now we resolve the remaining power:

jerarquía de paréntesis corchetes y llaves

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

prioridad en las operaciones matemáticas

operaciones combinadas con potencias

Combined operations with brackets and keys

JERARQUIA DE OPERACiONES 1

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:

operations combined with parentesis

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:

combined operations with powers

How to solve combined operations

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

solve combined operations

And finally we make the additions and subtractions:

exercises of operations combined with parentesis

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

combined operations with parentheses and brackets

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:

resolver paréntesis

How to solve combined operations with parentesis

Now it is possible to perform the division:

combined operations

And finally the additions and subtractions:

combined operations to solve

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:

How to solve parentesis brackets and keys in mathematical exercises

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:

operations combined with brackets

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

How to solve combined exercises

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

solve operations with parentesis

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:

How to do combined operations

We continue with the additions and subtractions within the parenthesis:

combined operations with powers and parentesis

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

operations combined with parentheses

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

operations combined with brackets and keys

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

operations with brackets and parentheses

Now, we add and subtract each of the parentheses:

exercises combined with parentesis

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

operations with parentesis

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

operations with parentheses and brackets

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

operations combined with power

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

How the combined operations are resolved

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

steps to solve combined operations

We first resolve the parenthesis:

operations combined with powers exercises solved

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

How to do combined operations with parentesis

We can also have two parentheses multiplying each other:

order to resolve operations combined with parentesis

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:

operations combined with parentesis exercises resolved

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:

examples of operations combined with parentesis

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

combined operations powers

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:

operaciones combinadas parentesis

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:

operations combined with parentesis and powers

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.

How to solve combined operations with brackets and keys

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

combined operations exercises to solve

We solve the parenthesis:

calculations combined with parentesis

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

How to solve combined calculations

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

operations with brackets and keys

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:

combined operations with brackets and parentheses

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

Now I am going to multiply the second parenthesis:

elimination of parentesis resolved exercises

We solve the parentheses:

combined operations exercises

Finally we remove parentheses and make the resulting sums:

exercises combined with brackets and parentheses

A last example to finish:

resolves step by step the operations of each parentesis

We resolve the parentheses inside the brackets:

How to solve a combined operation

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

how to solve combined operations

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:

examples of combined operations

Solution

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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:

operaciones combinadas con fracciones

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

ejercicios combinados de fracciones

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

operaciones combinadas con fracciones resueltas

In the end, we simplified the fraction.

Example 2:

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

ejercicios de operaciones combinadas con fracciones

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:

operaciones con fracciones combinadas

calculations combined with fractions

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

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

fracciones combinadas

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:

fracciones combinadas ejercicios resueltos

We keep breaking down the denominator:
operations combined with fractions exercises

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

combined fraction exercises

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:

fracciones combinadas ejercicios resueltos

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:

combined operations fractions

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:

how to solve combined operations with fractions

solved exercises of operations combined with 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:

operaciones con fracciones resueltas

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