Order of Operations with Parentheses

In previous articles, we have seen what is the order of operations for addition, subtraction, multiplication, and division and also the order we must follow when there are exponents.

When the exercise we need to solve includes parentheses, we always (always!) start with the operation contained within them.

Parentheses
Exponents and roots
Multiplications and divisions
Additions and subtractions

Reminder: when an exercise presents operations that have the same precedence, that is, multiplications and divisions or additions and subtractions, we will solve the exercise from left to right.

Detailed explanation

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Test yourself on parentheses!

$$ 10-(10-4):2= $$

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Below, we present to you some examples

Example 1

$4+(6:2)=$
In this exercise, we will start by solving the operation inside the parentheses, and then the rest:
$4+(6:2)=4+(3)= 4+3 = 7$

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Test your knowledge

Question 1

$$ (85+5):10= $$

Question 2

$$ 18-(3+3)= $$

Question 3

$8\times(5\times1)=$

Example 2

$5+8\cdot3-(8:4)=$
We'll start by solving the operation inside the parentheses:
$5+8\cdot3-2=$
Next, we continue with the multiplications:
$5+24-2=$
Finally, we add and subtract:
$5+24-2=27$

Example 3

$1+9\cdot15-(9:3)=$
We'll start by solving the operation inside the parentheses:
$1+9\cdot15-3=$
Next, we continue with the multiplications:
$1+135-3=$
Finally, we add and subtract:
$1+135-3=133$

Do you know what the answer is?

Question 1

$(7+2)\times(3+8)=$

Question 2

$187\times(8-5)=$

Question 3

$$ (29-4):5= $$

Example 4

$(21+3)\cdot2\operatorname{\cdot}4-(22:2)=$

We will start by solving the operation inside the parentheses:
$24\cdot2\operatorname{\cdot}4-11=$

Next, we continue with the multiplications:
$48\operatorname{\cdot}4-11=$

$192-11=$

Finally, we add and subtract:
$192-11=181$

Example 5

$(1+9)+(15\cdot8)-(8:2)=$

Let's start by solving the operation inside the parentheses:
$(10)+(120)-(4)=$

Finally, we add and subtract:
$10+120-4=126$

That is, the order in all exercises will be as follows:

Parentheses
Exponents and roots
Multiplications and divisions
Additions and subtractions

Reminder: when an exercise includes operations that have the same precedence, that is, multiplication and division or addition and subtraction, we will solve the exercise from left to right.

Examples and Exercises with Solutions on Order of Operations with Parentheses

examples.example_title

$(7+2)\times(3+8)=$

examples.explanation_title

Simplify this expression paying attention to the order of operations which states that exponentiation precedes multiplication and division before addition and subtraction and that parentheses precede all of them.

Therefore, let's first start by simplifying the expressions within parentheses, then we perform the multiplication between them:

$(7+2)\cdot(3+8)= \\ 9\cdot11=\\ 99$ Therefore, the correct answer is option B.

examples.solution_title

examples.example_title

$[(5-2):3-1]\times4=$

examples.explanation_title

In the order of operations, parentheses come before everything else.

We start by solving the inner parentheses in the subtraction operation:

$((3):3-1)\times4=$ We continue with the inner parentheses in the division operation and then subtraction:

$(1-1)\times4=$

We continue solving the subtraction exercise within parentheses and then multiply:

$0\times4=0$

examples.solution_title

$0$

examples.example_title

$12:3(1+1)=$

examples.explanation_title

First, we perform the operation inside the parentheses:

$12:3(2)$

When there is no mathematical operation between parentheses and a number, we assume it is a multiplication.

Therefore, we can also write the exercise like this:

$12:3\times2$

Here we solve from left to right:

$12:3\times2=4\times2=8$

examples.solution_title

examples.example_title

$9-6:(4\times3)-1=$

examples.explanation_title

We simplify this expression paying attention to the order of arithmetic operations which states that exponentiation precedes multiplication and division before addition and subtraction, and that parentheses precede all of them.

Therefore, we start by performing the multiplication within parentheses, then we carry out the division operation, and we finish by performing the subtraction operation:

$9-6:(4\cdot3)-1= \\ 9-6:12-1= \\ 9-0.5-1= \\ 7.5$

Therefore, the correct answer is option C.

examples.solution_title

7.5

examples.example_title

$(3\times5-15\times1)+3-2=$

examples.explanation_title

This simple rule is the order of operations which states that exponentiation precedes multiplication and division, which precede addition and subtraction, and that operations enclosed in parentheses precede all others,

Following the simple rule, multiplication comes before division and subtraction, therefore we calculate the values of the multiplications and then proceed with the operations of division and subtraction

$3\cdot5-15\cdot1+3-2= \\ 15-15+3-2= \\ 1$ Therefore, the correct answer is answer B.

examples.solution_title

$1$

Check your understanding

Question 1

$$ 10-(10-4):2= $$

Question 2

$$ (85+5):10= $$

Question 3

$4+(6+6:3)\cdot2=$

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