# Sequence of Operations

## Sequence of Operations

In arithmetic or algebra, a sequence of operations is a sequence of calculations to solve, including numbers and operation symbols, with or without the presence of parentheses or other grouping symbols.

The application of the properties of arithmetic operations and the order of operations leads to solving the operations in a sequence by following this hierarchy where possible:

• first, solve the calculations that appear in parentheses or other grouping symbols (P);
• simplify the calculations that include exponents (E);
• solve multiplication (M) and division (D) from left to right;
• and finally, solve addition (A) and subtraction (S) from left to right.

This hierarchy is known by the mnemonic acronym, PEMDAS.

### Example

Consider this sequence of operations: $$(23\space – 15)^2 × \dfrac{(12 + 35)}{4} + 10^3$$.  By following the hierarchy described above, we will calculate it like this:

• $$(23\space – 15)^2 × \dfrac{(12 + 35)}{4} + 10^3$$  =  $$8^2 × \dfrac{47}{4} + 10^3$$, by solving the parentheses (P);
• $$8^2 × \dfrac{47}{4} + 10^3$$  =  $$64 × \dfrac{47}{4} + 1000$$, by solving the exponents (E);
• $$64 × \dfrac{47}{4} + 1000$$  =  $$\dfrac{3008}{4} + 1000$$, by solving the multiplication (M);
• $$\dfrac{3008}{4} + 1000$$ = $$752 + 1000$$, by solving the division (D)
• $$752 + 1000$$ =   $$1752$$, by solving the addition (A).