Order of Operations
Look at â€œ4 + 5 Ã— 2â€. How do you know which to do first? Should you add 4 plus 5 and then
multiply by 2, or multiply first and then add?
Natural Order
There is a â€œnaturalâ€ order of operations to mathematics:
1. If thereâ€™s any parentheses ( ) then do them first.
2. Then do exponents x^{n} and powers 10^{3 }and radicals
.
3. Then do multiplication Ã— and division
Ã·.
4. And then do any addition + and subtraction  from left to right.
It may help to remember them as P.E.M.D.A.S. using their first letters.
Thus in the example 4 + 5 Ã— 2, we would multiply first.
Example:
8  4  1 = ?
Answer:
Since we have more than one subtraction, we do the left one first:
(8  4)  1 = 3
Parentheses
What if we wanted to tell someone to do the subtraction in a different order? Use parentheses
to change the natural order. Expressions in parentheses are always done first. Note: Singular is
parenthesis, and plural is parentheses.
Example:
8  (4  3) = ?
Answer:
The parentheses tell us to do â€œfour minus oneâ€ first:
8  (4  3) = 7
What about nested parentheses? Evaluate the innermost parentheses first.
Example:
( 8  (4 + 2))^{2} = ?
Answer:
( 8  6)^{2} = (2)^{2} = 4
Exponents
Suppose you have an expression with both subtraction and exponents. You should do the
exponent first (unless commanded otherwise by parentheses).
Example:
25  2^{4} = ?
Answer:
Since there is no parentheses ( ) we do â€œtwo to the fourth powerâ€ first:
25  16 = 9
Calculator Warning!
See your calculatorâ€™s manual to read about its order of operations. The cheapest ones simply
do everything lefttoright as you enter the operations. The better ones save the results of
multiplication and then add (or subtract) the products together. Try this on your calculator:
2 Ã—3 + 4 Ã—
5 = ?
A very basic calculator solves it like this, as you enter it from lefttoright:
2 Ã— 3 + 4 Ã—
5
6 + 4 Ã— 5
10 Ã— 5
50
A better calculator will follow the natural order of operations, and give a different answer even
though you pressed the same keys in the same order:
2 Ã— 3 + 4 Ã—
5
6 + 4 Ã— 5
6 + 20
26
So be sure to understand how your own calculator handles its order of operations!
By the way, most calculators that follow the natural order (multiplication first) also have
parentheses keys so you can change the order, if needed. If you see parentheses on a
calculator, thatâ€™s a good clue that it was designed to handle the natural order of operations. If in
doubt, use the parentheses keys â€œ(â€œ and â€œ)â€ to ensure it follows the order you want.
More Examples
If a = 4 and b = 3 compute the result:
Example 1:
a + 6  b = ?
Answer:
Add and subtract from left to right: 4 + 6  3 = 7
Example 2:
a / 2 Ã— b = ?
Answer:
Multiply and divide from left to right: 4 / 2 Ã—
3 = 6
Example 3:
b  a / 2 = ?
Answer:
The division comes first, then the subtraction: 3  4 / 2 = 1
Handling Fractions
One last complication: What about the expression
To figure this out, we must tell you there are always implied parentheses around the numerator
and denominator of a fraction. So first do addition and subtraction, and then do the division:
What about cascading fractions such as
? This can be either
or
which give
different results. The writer must be clear when putting it on paper! This is another great reason
to love parentheses.
Example:
Answer:
Use the implied parentheses around each fraction, then start with the innermost
and work your way out.
