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What is the standard interpretation of the order of operations for an expression involving some combination of grouping symbols, exponentiation, radicals, multiplication, division, addition, and subtraction?

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This question has been done to death here recently. See, for example, – Gerry Myerson May 2 '11 at 0:47
@Gerry: That's kind of the point—see this on meta. – Isaac May 2 '11 at 4:57
Ah. Might I suggest including that motivation in the statement of the question? – Gerry Myerson May 2 '11 at 5:02
up vote 10 down vote accepted

Any parts of an expression grouped with grouping symbols should be evaluated first, followed by exponents and radicals, then multiplication and division, then addition and subtraction.

Grouping symbols may include parentheses/brackets, such as $()$ $[]$ $\{\}$, and vincula (singular vinculum), such as the horizontal bar in a fraction or the horizontal bar extending over the contents of a radical.

Multiple exponentiations in sequence are evaluated right-to-left ($a^{b^c}=a^{(b^c)}$, not $(a^b)^c=a^{bc}$).

It is commonly taught, though not necessarily standard, that ungrouped multiplication and division (or, similarly, addition and subtraction) should be evaluated from left to right. (The mnemonics PEMDAS and BEDMAS sometimes give students the idea that multiplication and division [or similarly, addition and subtraction] are evaluated in separate steps, rather than together at one step.)

Implied multiplication (multiplication indicated by juxtaposition rather than an actual multiplication symbol) and the use of a $/$ to indicate division often cause ambiguity (or at least difficulty in proper interpretation), as evidenced by the $48/2(9+3)$ or $48÷2(9+3)$ meme. This is exacerbated by the existence of calculators (notably the obsolete Texas Instruments TI-81 and TI-85), which (at least in some instances) treated the $/$ division symbol as if it were a vinculum, grouping everything after it.

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Better would be to refer to this paper "Order of operations" and other oddities in school mathematics by Hung-Hsi Wu, a mathematician who has given much thought to mathematics education. – Bill Dubuque May 2 '11 at 23:23
@Bill: If you think that's a better answer, feel free to post it as an answer. – Isaac Jul 4 '11 at 0:05

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