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1.4 | Order of Operations and PEMDAS Rule |

In these revision notes for Order of Operations and PEMDAS Rule, we cover the following key points:

- What is the order of operations in arithmetic expressions that contain only additions and divisions?
- What are the order of operations in arithmetic expressions that contain only multiplications and divisions.
- What are the order of priority in expressions containing powers and the four basic operations?
- How do brackets affect the order of priority of arithmetic operations?
- What is the PEMDAS Rule?
- What is the order of priority in expressions containing various types of brackets?
- What is the sign of result in operations with negative numbers?
- How do you calculate the result of powers of negative numbers?

**Addition and subtraction are arithmetic operations of the same order of priority**. This means that if an arithmetic expression contains only additions and subtractions, we start operations at the left and work to the right, regardless of the operation involved.

**Multiplication and division have the same order of priority**; hence, in an arithmetic expression containing only multiplications and divisions, the operations are done from left to right regardless of the type of operation that comes first.

**Multiplications and divisions are of the same order of priority with each other but of a higher order of priority than additions and subtractions**. Therefore, if an arithmetic expression contains all four basic operations, we start the calculations by completing the multiplications and divisions first, regardless of their position in the expression. Then, after completing multiplications and divisions, we continue with the rest (additions and subtractions) working from left to right.

Expressions containing exponents are those algebraic expressions in which at least one of the numbers is raised by a given power. In these expressions, **powers are calculated first, as they have a higher order of priority than multiplications and divisions (and obviously of addition and subtractions)**.

**Brackets** are used in an arithmetic expression to highlight the part of expression enclosed inside them, i.e. to indicate the part of expression that is to be solved first, regardless of the operations involved. In this case, we forget the rest of the equation and focus on the content inside the brackets. After removing brackets, we then continue with the rest of expression.

We can use the **PEMDAS Rule** to make it easier to remember the order of operations in expressions containing all operations mentioned above as well as brackets. PEMDAS is the acronym formed using the first letter of the rules discussed so far when put in priority order:

P → Parenthesis (brackets)

E → Exponents

M → Multiplication

D → Division

A → Addition

S → Subtraction

E → Exponents

M → Multiplication

D → Division

A → Addition

S → Subtraction

An arithmetic expression may be more complicated and require other types of parenthesis I addition to the **round** ones, ( ). Two other types of brackets that are commonly used in arithmetic expressions are the **square** brackets, [ ] and **curled** ones { }. When calculating equations which contain different types of brackets, the priority order is: Round brackets are calculated first, then square brackets and finally the curled ones.

The sign rules for each operation are:

For addition, we have the following rules:,/p>

- positive + positive = positive
- negative + negative = negative
- positive + negative = their difference with the sign of the number that is farther from origin

Since subtraction is the inverse operation of addition, we turn the subtraction into addition and at the same time, we change the sign of the second number for compensation. Then, we apply the same rules as in addition.

Multiplication. There are two basic rules in multiplication:

- The product of two factors of the same sign is positive
- The product of two factors of opposite signs is negative

Since division is the inverse operation of multiplication, the same rules for multiplication are applied.

When negative numbers are raised by a certain power, we apply the same rules as used in multiplication. Thus, if we have a negative number raised by an even power, the result is positive as we can do the operations two by two. On the other hand, when a negative number is raised by an odd power, the result is negative. This is due to the fact that when exponents are grouped two by two they give a positive number which, when multiplied with the only negative number left, becomes negative.

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