Order of operations
Conventional order

The order of operations, that is, the order in which the operations in an expression are usually performed, results from a convention adopted throughout mathematics, science, technology and many computer programming languages. It is summarized as:

This means that to evaluate an expression, one first evaluates any sub-expression inside parentheses, working inside to outside if there is more than one set. Whether inside parenthesis or not, the operation that is higher in the above list should be applied first. Operations of the same precedence are conventionally evaluated from left to right.

If each division is replaced with multiplication by the reciprocal (multiplicative inverse) then the associative and commutative laws of multiplication allow the factors in each term to be multiplied together in any order. Sometimes multiplication and division are given equal precedence, or sometimes multiplication is given higher precedence than division; see § Mixed division and multiplication below. If each subtraction is replaced with addition of the opposite (additive inverse), then the associative and commutative laws of addition allow terms to be added in any order.

The root symbol √ is traditionally prolongated by a bar (called vinculum) over the radicand (this avoids the need for parentheses around the radicand). Other functions use parentheses around the input to avoid ambiguity. The parentheses can be omitted if the input is a single numerical variable or constant, as in the case of sin x = sin(x) and sin π = sin(π). Traditionally this convention extends to monomials; thus, sin 3 x = sin(3 x) and even sin 1 / 2 xy = sin(xy /2), but sin x + y = sin(x) + y, because x + y is not a monomial. However, this convention is not universally understood, and some authors prefer explicit parentheses. Some calculators and programming languages require parentheses around function inputs, some do not.

Symbols of grouping can be used to override the usual order of operations. Grouped symbols can be treated as a single expression. Symbols of grouping can be removed using the associative and distributive laws, also they can be removed if the expression inside the symbol of grouping is sufficiently simplified so no ambiguity results from their removal.
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