When we have a number of operations together, like adding, subtracting, multiplying, dividing and using exponents, we need to know what order to do them. Don’t worry why we 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 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 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 | ||||
Here are some hints:
- Do the math left to right, like you’re reading a book, but you must pay careful attention and perform some operations before others.
- With embedded parentheses (parentheses within parenthesis), perform operations inside out; do the math on the inside before the outside.
- If you have an absolute value | | sign (which means take the positive value only of a number inside), treat it like a parenthesis – so do that math first.
- With 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.
- When you start working with variables when Solving Algebraic Equations, 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, 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}$ | First subtract $ 3$ from $ 5$, since they are in parentheses; this equals $ 2$. Then square $ 2$, since exponents are next; we get $ 4$. Then multiply $ 4$ by $ 6$ since multiplication is before subtraction. Then 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}$ | Since we have to perform multiplication or division from left to right, divide $ 10$ by $ 2$ first to get $ 5$. Then multiply $ 5$ by $ 6$ to get $ 30$, and finally subtract $ 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. Perform the operations from the inside out, so first subtract $ -2$ from $ -4$ to get $ -6$. Then multiply $ -6$ by $ -1$ to get $ 6$. Then square $ 6$ to get $ 36$, and add it to $ 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; the absolute value of $ -2$ is $ 2$. Square $ 3$ to get $ 9$, do the multiplication, and finally the subtraction, from left to right. |
| $ \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; perform 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!