Difference between revisions of "Order of operations"
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| − | The '''order of operations''' is a [[mathematical convention]] for [[arithmetic]] computation. The order | + | The '''order of operations''' is a [[mathematical convention]] for [[arithmetic]] computation. The below list indicates the order to do arithmetic, from top to bottom. |
| − | ==See | + | * [[Parentheses]] -- when there are multiple operations and parentheses within the parentheses, treat the section inside as a separate expression and do the order of operations within said parentheses |
| − | * [[ | + | * [[Exponent|Exponents]] |
| + | * [[Multiplication]] and [[Division]] -- done from left to right | ||
| + | * [[Addition]] and [[Subtraction]] -- done from left to right | ||
| + | |||
| + | The order of operations is usually summarized by the acronym PEMDAS. An AoPS mnemonic you can use to remember the order of operations is "Please Evaluate, My Dear AoPS Students". | ||
| + | |||
| + | Sometimes, properties like the Communicative Property, Associative Property, and [[Distributive property|Distributive Property]] can be applied. | ||
| + | |||
| + | == Video on Order Of Operations == | ||
| + | [https://youtu.be/Ti9nA3Tn_CE Order of Operations] | ||
| + | |||
| + | ==Examples== | ||
| + | |||
| + | ===Example 1=== | ||
| + | |||
| + | Consider the expression <math>2^8-(3+5)</math>. First, we do addition first because it is in parentheses. | ||
| + | <cmath>2^8 - 8</cmath> | ||
| + | Next, we evaluate <math>2^8</math> because exponents go next. | ||
| + | <cmath>256-8</cmath> | ||
| + | Afterward, there are no multiplication and division, so we can move on to addition and subtraction. | ||
| + | <cmath>248</cmath> | ||
| + | Therefore, <math>2^8 - (3+5) = \boxed{248}</math>. | ||
| + | |||
| + | ===Example 2=== | ||
| + | |||
| + | Consider the expression <math>14(4\times 7-58)-1</math>. The operations in the parentheses go first, but there are subtraction and multiplication both involved. Within the parentheses, the multiplication goes first, then the subtraction. | ||
| + | <cmath>14(28 - 58) - 1</cmath> | ||
| + | <cmath>14(-30) - 1</cmath> | ||
| + | Afterward, we do multiplication, then subtraction. | ||
| + | <cmath>-420-1</cmath> | ||
| + | <cmath>-421</cmath> | ||
| + | Therefore, <math>14(4\times 7-58)-1 = \boxed{-421}</math>. | ||
| + | |||
| + | ==Problems== | ||
| + | |||
| + | * Problems on [https://artofproblemsolving.com/alcumus/ Alcumus] | ||
| + | ** Order of Operations | ||
| + | * [[2017 AMC 10A Problems/Problem 1]] | ||
| + | |||
| + | ==See Also== | ||
| + | * [[Algebra]] | ||
Latest revision as of 12:19, 5 March 2022
The order of operations is a mathematical convention for arithmetic computation. The below list indicates the order to do arithmetic, from top to bottom.
- Parentheses -- when there are multiple operations and parentheses within the parentheses, treat the section inside as a separate expression and do the order of operations within said parentheses
- Exponents
- Multiplication and Division -- done from left to right
- Addition and Subtraction -- done from left to right
The order of operations is usually summarized by the acronym PEMDAS. An AoPS mnemonic you can use to remember the order of operations is "Please Evaluate, My Dear AoPS Students".
Sometimes, properties like the Communicative Property, Associative Property, and Distributive Property can be applied.
Contents
Video on Order Of Operations
Examples
Example 1
Consider the expression 
. First, we do addition first because it is in parentheses.
![\[2^8 - 8\]](http://latex.artofproblemsolving.com/6/6/c/66c2261e10837dce39bec8511e46a4f9cbef4cda.png)
Next, we evaluate 
because exponents go next.
![\[256-8\]](http://latex.artofproblemsolving.com/b/5/7/b5728ef5db4ccf3e774bd2a9ef6aa6cf6950d341.png)
Afterward, there are no multiplication and division, so we can move on to addition and subtraction.
![\[248\]](http://latex.artofproblemsolving.com/e/c/b/ecbab2d064fe6ffdc9353fecf1b13e8aa7b58276.png)
Therefore, 
.
Example 2
Consider the expression 
. The operations in the parentheses go first, but there are subtraction and multiplication both involved. Within the parentheses, the multiplication goes first, then the subtraction.
![\[14(28 - 58) - 1\]](http://latex.artofproblemsolving.com/8/0/4/8042fed24a460bbd87c59d14662af44b41cae557.png)
![\[14(-30) - 1\]](http://latex.artofproblemsolving.com/8/f/7/8f7e1ee6df49f524217aa4fa2754e51968cfe2b6.png)
Afterward, we do multiplication, then subtraction.
![\[-420-1\]](http://latex.artofproblemsolving.com/5/4/2/542eca4f53aff72a8328679802b2bba20f1658e6.png)
![\[-421\]](http://latex.artofproblemsolving.com/f/3/4/f34b5e53627e4ea817a268510d004245b8994585.png)
Therefore, 
.
Problems
- Problems on Alcumus
- Order of Operations
- 2017 AMC 10A Problems/Problem 1
