Explanation:
A math expression is a collection of symbols. A set of rules for their interpretation is needed in order for anyone who uses the expression to get the same result. That set of rules is called the "order of operations."
Generally, when we talk about the "order of operations", we're referring to a short set of rules that are often referred to by a 6-character mnemonic term, usually BIDMAS or PEMDAS. (The former tends to be used in the UK; the latter in the US. There may be other variations.) These mnemonic letters help remind one of the precedence of operations with respect to each other.
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B/P -- Brackets/Parentheses -- means the highest precedence is given to any expression in parentheses. When parentheses are nested, the innermost expression is to be evaluated first.
I/E -- Indices/Exponents -- means exponentiation has the highest precedence after parentheses. [Where functions are used/referenced, often function evaluation is given a higher precedence than exponentiation.] Exponents are evaluated right to left, so a^b^c means a^(b^c).
DM/MD -- Multiplication and Division -- means these operations have equal precedence, and are performed in the order encountered, left to right. This means special attention needs to be given to denominators that are a product: a/bc means (a/b)c, not a/(bc). If the latter is intended, then the parentheses are required. (They can be useful in the former case to make sure the meaning is understood.) Similarly, exponents that are an expression need parentheses: a^b/c means (a^b)/c, for example.
The operations of multiplication and division have equal precedence. Some folks like to interpret DM to mean division is done before multiplication. That is not the intent. These operations are performed in the order encountered.
AS -- Addition and Subtraction -- means these operations have equal precedence, and are performed in the order encountered, left to right.
Commutative and associative properties
In evaluating expressions, the above rules of precedence apply. However, also applicable are the properties of these operations. Multiplication has commutative and associative properties, as well as inverse properties, allowing the order of multiplication (and division) to be changed.
abc/de/f = (abce)/(df) = (a/d)(b/f)ce . . . . if you like
Similarly, addition has commutative, associative, and inverse properties that can be used. Using these properties, we can rewrite expressions to equivalents:
a +b -c +d -e = a+b+d -c-e
and the distributive property lets us even go further:
= (a+b+d) -(c+e)
Additional comments and examples
You may run across some questions intended to test your understanding of the order of operations. Here are a couple:
6 ÷2(1 + 2) = 6÷2·3 = 3·3 = 9
9 -3 ÷1/3 +1 = 9 -3/3 +1 = 9 -1 +1 = 8 +1 = 9 . . . as written
Often, the 1/3 is intended to be a fraction. It should be written so that it is not confused with a division operation.
9 -3 ÷(1/3) +1 = 9 -3×3 +1 = 9 -9 +1 = 0 +1 = 1 . . . as often intended
You will note that both ÷ and / are used to signify division. Sometimes "over" is also used for that purpose. The order of operations makes no differentiation between these forms of the division operator (see the second example immediately above). Some textbook authors do assign different properties to these operators. Beware.
