# Order of Operations

## Types of Combined Operations

### 1. Operations without Parentheses

**1.1Additions and subtractions. **

9 − 7 + 5 + 2 − 6 + 8 − 4 =

Starting from the left, carry out operations as they appear.

= 9 − 7 + 5 + 2 − 6 + 8 − 4 = 7

**1.2 Additions, subtractions and multiplications. **

3 · 2 − 5 + 4 · 3 − 8 + 5 · 2 =

Carry out multiplications first, as they are higher priority.

= 6 − 5 + 12 − 8 + 10 =

Then, perform the** additions and subtractions**.

= 6 − 5 + 12 − 8 + 10 = 15

**1.3 Additions, subtractions , multiplications and divisions. **

10 : 2 + 5 · 3 + 4 − 5 · 2 − 8 + 4 · 2 − 16 : 4 =

Perform the **multiplications** and **divisions** in the order in which they appear, as the two operations have the same priority.

= 5 + 15 + 4 − 10 − 8 + 8 − 4 =

Then, perform the** additions and subtractions**.

= 5 + 15 + 4 − 10 − 8 + 8 − 4 = 10

**1.4 Additions, subtractions, multiplications, divisions and exponents. **

2^{3 } + 10 : 2 + 5 · 3 + 4 − 5 · 2 − 8 + 4 · 2^{2} − 16 : 4 =

First, the **powers** have the highest priority.

= 8 + 10 : 2 + 5 · 3 + 4 − 5 · 2 − 8 + 4 · 4 − 16 : 4 =

Continue with the **multiplications and divisions**.

= 8 + 5 + 15 + 4 − 10 − 8 + 16 − 4 =

Then, perform the** additions and subtractions**.

= 26

### 2. ** Operations with Parentheses **

(15 − 4) + 3 − (12 − 5 · 2) + (5 + 16 : 4) −5 + (10 − 2^{3})=

First, calculate the values** contained within the parentheses**.

= [15 − (8 − 5 )] · [5 + (6 − 4 )] − 3 + (8 − 6 ) =

Then, **eliminate the parentheses** making the operations.

= 11 + 3 − 2 + 9 − 5 + 2 = 18

### 3. Operations with Parentheses and Brackets

[15 − (2^{3} − 10 : 2 )] · [5 + (3 · 2 − 4 )] − 3 + (8 − 2 · 3 ) =

First, operate with the **exponents, multiplications and divisions of the parentheses**.

= (15 − (8 − 5 )] · [5 + (6 − 4 )] − 3 + (8 − 6 ) =

Then, carry out the** addition and subtraction of the parentheses**.

= [15 − 3] · [5 + 2 ] − 3 + 2 =

Instead of inserting brackets they are already in place:

= (15 − 3) · (5 + 2) − 3 + 2=

Operate within the parentheses.

= 12 · 7 − 3 + 2

**Multiply**.

= 84 − 3 + 2=

**Subtract and add.**

= 83