The Order of Operations has you evaluate arithmetic expressions starting with contents of parentheses. Within that, you start with exponentiation. First, you evaluate (-4)^2 = 16.
Now, you have a sum in parentheses. Evaluate that next.
... (-8 +16 -6) = 2
Now, you have multiplication and division that need to be evaluated (left to right).
... -12÷3×2 = -4×2 = -8
Finally, you have addition:
... -8 + 2 = -6
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Whenever you're in doubt, you can type your expression into a Google or Bing search box. Those calculators reliably apply the order of operations to what you type there, so be sure you have all necessary parentheses in place.
Order of Operations
There are several mnemonics in use to try to help you remember the Order of Operations. A couple of them are ... BIDMAS and PEMDAS. The latter sometimes is augmented by assigning words to the letters: Please Excuse My Dear Aunt Sally.
These have the purpose of reminding you the priority order of operations in an arithmetic or algebraic expression. (Some operations are left out--such as function evaluation--and some need "interpretation". A surd, for example, represents a fractional exponent.)
Here's what these mnemonics mean.
B or P — Brackets or Parentheses. Contents of parentheses are evaluated first. Use the order of operations to do that.
I or E — Indices or Exponents. Raising to a power has the highest priority. In the absence of parentheses, powers are evaluated right to left: a^b^c means a^(b^c), for example. This can get you in trouble if there are arithmetic operations in the exponent. For example x^1/2 means (x^1)/2 = x/2 because the power is evaluated before the division. I advise parenthses around any exponent that is not a single letter or number.
If you want to raise a negative number to a power, it must be enclosed in parentheses: (-1)^2 = (-1)×(-1) = 1; -1^2 = -(1×1) = -1. √3x means (√3)x because √3 = 3^(1/2), and the power is evaluated before the multiplication. If you want the root of a fraction or product, parentheses are needed.
D or M
M or D — Multiplication and division. It is important to note that these have the same priority, so are executed left to right in whatever order they are found. Some students erroneously attach significance to the ordering of these letters, so do multiplication before division or vice-versa. These two operations are simply done in the order they appear—neither having priority over the other. Again, trouble comes when the operands include other operations: 1/2·3 ≠ 1/(2·3). I generally advise parentheses around all numerators and denominators—especially if more than one operand is involved.
A or A
S or S — Addition and subtraction. As with multiplication and division, these operations have the same priority, so are executed in order left to right.
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The commutative and associative properties of multiplication and addition can be used to simplify problems. That is, the order of multiplication operands can be altered, as can the order of addition operands.* While that is generally true of expressions evaluated by humans with a calculator, expressions evaluated by computer have other considerations (related to truncation, rounding, and representation of numbers) that come into play. Computers have their own rules, and neither they nor calculators are created equal in regard to order of operations. Beware.
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* Remember that subtraction is simply addition of a negative number, division is simply multiplication by a reciprocal. Taking these into account, you can rearrange subtraction and/or division to your liking, as well. Make sure that changing the order does not change the meaning of what you're doing.
Examples: 12×3/2 = 12/2×3 = 18
... 3 + 6 - 3 = 3 - 3 + 6 = 6
Reminder: when working "order of operations" problems, be sure to adhere to the prescribed order. When evaluating an expression for your own use, commutative and associative properties can be your friend.
