Order of Operations

When you work with equations to simplify them, you work in specific ways. Generally, you perform multiplication and division first, working from left to right. Then you perform addition and subtraction, working from left to right. Consider these expressions.

3 × 3 – 2 = 9 – 2 = 7

3 × 3 – 2 – 2 × 2 = 9 – 2 – 4 = 7 – 4 = 3

3 × 3 – 2 – 2 ÷ 2 = 9 – 2 – 1 = 7 – 1 = 6

If you find the multiplication and division operations and perform those first, you are usually on safe ground. Still, to make it so that the order of operations is easier to understand, you can employ parentheses:

(3 × 3) – 2 = 9 – 2 = 7

(3 × 3) – 2 – (2 × 2) = 9 – 2 – 4 = 7 – 4 = 3

(3 × 3) – 2 – (2 ÷ 2) = 9 – 2 – 1 = 7 – 1 = 6

Parentheses override the standard order of operations. When you work with parentheses, you perform the operations in the innermost parentheses first and then proceed outward. After calculating the innermost grouping symbols, you then simplify the exponential expressions. After that, you carry out operations according to the usual order. Here is an example:

2³ + 33 × (3 + 4(2 + 2)) – (3 × 3)(3)

= 2³ + 33 × (3 + 4(4)) – (9)(3)

= 2³ + 33 × (3 + 16) – (27)

= 2³ + 33 × (19) – (27)

= 8 + 33 × (19) – (27)

= 8 + (33 × 19) – 27

= 8 + 627 – 27

= 635 – 27

= 608