Usually we have a number of operations (operations are like adding, subtracting, multiplying, dividing and using exponents) together, we need to know what order to do them (which ones to do first). Don’t worry whywe perform the operations in these orders; this is just something you have to memorize and not worry about.
This section is very important, since you’ll use this all through Calculus and college. Another thing that’s quite interesting with learning the order of operations is that your calculator will automatically perform the correct order if you type in the numbers correctly (including with parentheses).
I like to use the mnemonic “Please Excuse My Dear Aunt Sally” or PEMDAS to remember the order. The order is Parentheses, Exponents, Multiplication and Division (either one, in order from left to right), and then Addition and Subtraction (either one, in order from left to right) – PEMDAS.
My Dear Aunt Sally
PEMDAS stands for:
PLEASE | EXCUSE | MY | DEAR | AUNT | SALLY |
Parentheses | Exponents | Multiplication | Division | Addition | Subtraction |
Left to right, perform either multiplication OR division | Left to right, perform either addition OR subtraction |
So you’ll do the math left to right, like you’re reading a book, but you must pay careful attention and perform some operations before others.
Note that with embedded parentheses (parentheses within parenthesis), you always perform operations inside out; do the math on the inside before the outside.
Note that if you have an absolute value | | sign (which means take the positive value only of a number inside), you treat it like a parenthesis – so do that math first.
Also, if you have a fraction, and there are operations in the numerator or denominator, it’s almost like there are parentheses around them, so you do those first.
Also note that when you start working with variables in Algebra, you have to do the same order of operations.
Here are some examples:
Order of Operation Example | Explanation |
\(\begin{array}{c}\color{#800000}{{5+8\div 2}}\\5+4\\9\end{array}\) | Since we have to perform multiplication and division before addition and subtraction, we have to first divide 8 by 2, which is 4, and then add it to 5, which is 9. |
\(\begin{array}{c}\color{#800000}{{4-{{{\left( {5-3} \right)}}^{2}}\times 6}}\\4-{{\left( 2 \right)}^{2}}\times 6\\4-4\times 6\\4-24\\-20\end{array}\) | We have to first subtract the 3 from the 5, since they are in the parentheses; this equals 2. Then we have to square 2, since we have to perform exponents next; we get 4. Then we have to multiply 4 by 6 since we need to do multiplication before subtraction. Then we have to subtract 24 from 4 to get –20. |
\(\begin{array}{c}\color{#800000}{{10\div 2\times 6-3}}\\5\times 6-3\\30-3\\27\end{array}\) | This one is a little tricky. Since we have to perform multiplication or division from left to right, we need to divide 10 by 2 first to get 5. Then we have to multiply the 5 by 6 to get 30, and finally subtract the 3 from 30 to get 27. |
\(\displaystyle \begin{array}{c}\color{#800000}{{5+{{{\left[ {-1\,\left( {-4-2} \right)} \right]}}^{2}}}}\\5+{{\left[ {-1\left( {-6} \right)} \right]}^{2}}\\5+{{[6]}^{2}}\\5+36\\41\end{array}\) | Sometimes we have two sets of parentheses in problems, where we use parentheses inside brackets. We have to perform the operations from the inside out, so we first have to subtract the –2 from the –4 to get –6. Then we have to multiply it by –1 to get 6. Then we have to square the 6 to get 36, and add it to the 5 to get 41. |
\(\displaystyle \begin{array}{c}\color{#800000}{{4-\left| {8-10} \right|\cdot {{3}^{2}}-1}}\\4-\left| {-2} \right|\cdot {{3}^{2}}-1\\4-\left( 2 \right)\cdot \left( 9 \right)-1\\4-18-1\\-15\end{array}\) | Treat the absolute value sign as a parenthesis, so do that math first. Note that the absolute value of –2 is 2. Then we square the 3 to get 9, do the multiplication, and finally the subtraction. |
\(\displaystyle \frac{{100-25}}{{4+3\times 2}}+4\times 10\)
\(\displaystyle \frac{{\left( {100-25} \right)}}{{\left( {4+3\times 2} \right)}}+40\) \(\displaystyle \frac{{75}}{{10}}+40\) \(\displaystyle \begin{array}{c}7.5+40\\47.5\end{array}\) |
Treat the operations in the numerator and denominator as having parentheses; we’ll then do the multiplication and division operations before the addition and subtraction. |
Learn these rules, and practice, practice, practice!
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On to Introduction to Statistics and Probability – you are ready!