When attempting to evaluate an expression like the one above, where do
you begin??? Can you begin anywhere? If you do, you may find that someone
a different position and received a different answer. No good! You can't
have two or more conflicting results to the same expression. Some may
be tempted to just start on the left and work towards the right. Unfortunately,
you will get a result, just not the intended one.
In order to make sure that everyone gets the same result, an particular
precedence has been given to some of the operations. In other words, some
of the operations must be evaluated first. Others have to wait. Once this
"order of operations" has been mastered, evaluating these expressions should
be easy,. However, you'll quickly find you need to pay close attention at
every step, for mistakes inevitably come. So what is this order???
Grouping symbols are given first priority in any mathematical expression.
You must begin evaluating what is inside the grouping symbols before
anything else in the expression. Grouping symbols include BOTH parentheses
( ) and square brackets [ ]. That's right, brackets are also included.
So those of you that remember "Please Excuse My Dear Aunt Sally",
better do something about that first word!
Practical Note: In the expression above, the portion in brackets would
be the first part of the expression to evaluate.
Next on the priority list is exponents. If there are no more grouping
symbols in your expression, you must go to your exponents next. Usually
this takes place naturally, but you should make a point of evaluating
them all after the grouping symbols.
Practical Note: In the expression above, we start in the brackets.
Once in the brackets, since there are no additional grouping symbols,
we start evaluating with exponents - 4 squared.
This is where we start to make mistakes. Many of the acronyms and phrases
people recite in order to remember the order of operations mislead them
at this step. Multiplication and division both have the same level of
precedence, one is not given priority over the other. So how do I know
which to do first? When it comes to multiplication and division, since
they have the same precedence we start on the left and work toward the
right. Why in that direction? Well, because we read from left to right.
Practical Note: In the above expression, after everything in the brackets
is evaluated, we would then, since there wouldn't be any exponents,
evaluate the division, then the multiplication, for the division comes
first from left to right.
Addition and subtraction are at the bottom of the priority list. Similar
to multiplication and division, addition and subtraction have the same
level of precedence. Therefore, they are both evaluated from left to
right as well.
Practical Note: In the expression above, after the brackets, multiplication,
and division have been evaluated, we would then evaluate the addition
and subtraction from left to right.
||1. Since there are no grouping symbols or exponents,
we evaluate multiplication and division, from left to right. Therefore,
to right, the first multiplication/division problem we see is 10 divided
by 2. That is 5.
2. There is only one more multiplication/division problem - 5 times
5. That is 25.
3. Since there is only addition and subtraction left, we evaluate
them from left to right - first starting with 5 plus 25. That is
4. Only one thing left to evaluate - 30 minus 2. That, is 28.
||1. We must first start in the grouping symbols. Once in the grouping
symbols, we start at the top of the priority list again. Any more
grouping symbols? Nope. Any exponents? Nope. Any multiplication or
division? Yes! Since there is only one division problem and no multiplication,
we start with that - 16 divided by 2. That is 8.
2. Still in the grouping symbols, we have only one problem left
- 10 minus 8. That is 2.
3. Since there is nothing left to evaluate in the grouping symbols,
and because there are no more grouping symbols left in the expression,
we must now look for exponents. There is only one expression with
- 4 squared.
That is 16.
4. With no grouping symbols or exponents, we are to evaluate multiplication
and division from left to right. The first multiplication/division
problem from the left is 16 divided by 2. That is 8.
5. The next multiplication/division problem is 3 times 4. That
6. Now, there is only addition and subtraction. Working from left
to right we have the first problem - 20 minus 8. That is 12.
7. The last problem - 12 plus 12. That is 24.
Order of Operations Quick Reference Diagram