Order of Operations with Parentheses

Parentheses in advanced Order of Operations - Examples, Exercises and Solutions
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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.

  1. Parentheses
  2. Exponents and roots
  3. Multiplications and divisions
  4. 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.

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Test yourself on parentheses in advanced order of operations!

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

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Below, we present to you some examples

Example 1

4+(6:2)=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=74+(6:2)=4+(3)= 4+3 = 7

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Test your knowledge
Question 1

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

Question 2

\( 8\times(5\times1)= \)

Question 3

\( (2+1\times2)^2= \)

Example 2

5+8⋅3−(8:4)=5+8\cdot3-(8:4)=
We'll start by solving the operation inside the parentheses:
5+8⋅3−2=5+8\cdot3-2=
Next, we continue with the multiplications:
5+24−2=5+24-2=
Finally, we add and subtract:
5+24−2=275+24-2=27

Example 3

1+9⋅15−(9:3)=1+9\cdot15-(9:3)=
We'll start by solving the operation inside the parentheses:
1+9⋅15−3=1+9\cdot15-3=
Next, we continue with the multiplications:
1+135−3=1+135-3=
Finally, we add and subtract:
1+135−3=1331+135-3=133

Do you know what the answer is?
Question 1

\( 18-(3+3)= \)

Question 2

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

Question 3

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

Example 4

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

We will start by solving the operation inside the parentheses:
24⋅2⋅⁡4−11= 24\cdot2\operatorname{\cdot}4-11=

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

192−11= 192-11=

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

Example 5

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

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

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

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

  1. Parentheses
  2. Exponents and roots
  3. Multiplications and divisions
  4. 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

Exercise #1

10−(10−4):2= 10-(10-4):2=

Video Solution

Step-by-Step Solution

Let's simplify this expression while following the order of operations which states that exponents come before multiplication and division, which come before addition and subtraction, and that parentheses come before all of them,

We will start by simplifying the expression inside the parentheses and calculate the result of the subtraction within them, then - since division comes before subtraction, we will first perform the division operation and then perform the subtraction operation:

10−(10−4):2=10−6:2=10−3=7 10-(10-4):2= \\ 10-6:2= \\ 10-3= \\ 7

Therefore, the correct answer is answer B.

Answer

7

Exercise #2

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

Video Solution

Step-by-Step Solution

According to the order of operations rules, we must first solve the expression within the parentheses:

85+5=90 85+5=90

We should obtain the following expression:

90:10=9 90:10=9

Answer

9

Exercise #3

18−(3+3)= 18-(3+3)=

Video Solution

Step-by-Step Solution

Let's simplify this expression while following the order of operations which states that exponents come before multiplication and division, which come before addition and subtraction, and that parentheses come before all of them,

therefore we'll start by simplifying the expression inside the parentheses and perform the addition within them, then we'll perform the subtraction operation that applies to the expression in parentheses:

18−(3+3)=18−6=12 18-(3+3)= \\ 18-6= \\ 12 Therefore the correct answer is answer D.

Answer

12

Exercise #4

8×(5×1)= 8\times(5\times1)=

Video Solution

Step-by-Step Solution

According to the order of operations, we first solve the expression in parentheses:

5×1=5 5\times1=5

Now we multiply:

8×5=40 8\times5=40

Answer

40

Exercise #5

(2+1×2)2= (2+1\times2)^2=

Video Solution

Step-by-Step Solution

Let's solve the expression (2+1×2)2 (2+1\times2)^2 step-by-step, adhering to the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

Firstly, handle the expression inside the parentheses (2+1×2) (2+1\times2) :

  • Within the parentheses, according to PEMDAS, we first perform the multiplication 1×21\times2 which equals 22.
  • Now, the expression inside the parentheses becomes (2+2) (2+2) .
  • Next, perform the addition: 2+2=42+2=4.

Now the expression simplifies to 424^2.

Second, handle the exponent:

  • Calculate the square of 4: 42=164^2 = 16.

Thus, the final answer is 1616.

Answer

16

Check your understanding
Question 1

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

Question 2

Solve the following:

\( 18-(3+3)= \)

Question 3

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

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