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Order of Operations

The Order of Operations tells us when to do each step in a math problem. There are some problems that do not need to use the order of operations (such as 1+2+3), but in Algebra, it is rare you see such simplistic operations. Luckily, the rules for the orders of operations are easy.

Orders of Operations
1. Do all the calculations for operations inside of parenthesis.
2. Compute any exponents and radicals.
3. Working left to right, perform all multiplication and division.
4. Working left to right, perform addition and subtraction.

That's all you have to remember. Let's try a few.


Solve: 20 + (2 * 5) + 2 - 2²
Step Rule Result
1 Do all the calculations for operations inside of parenthesis. 20 + (10) + 2 - 2²
2 Compute any exponents and radicals 20 + (10) + 2 - 4
3 Working left to right, perform all multiplication and division. 20 + 10 + 2 - 4
4 Working left to right, perform addition and subtraction. 20+10+2-4 = 28


Solve: 1 + (3 + (10/2) - 5) * 3
Step Rule Result
1 Do all the calculations for operations inside of parenthesis. 1 + (3 + (5) - 5) * 3
1 + (3) * 3
2 Compute any exponents and radicals 1 + 3 * 3
3 Working left to right, perform all multiplication and division. 1 + 9
4 Working left to right, perform addition and subtraction. 1 + 9 = 10

Solve: (3 + (2 * 5 + (3² + 2) - 9 )) * (2³ - 2)
Step Rule Result
1 Do all the calculations for operations inside of parenthesis.
Note: While inside of each inner-most parenthesis, you must perform each other step below to determine the values for inside those parenthesis!
(3 + (2 * 5 + (3² + 2) - 9 )) * (2³ - 2)
2 Compute any exponents and radicals (3 + (2 * 5 + (9+2) - 9)) * (8-2)
3 We still have inner-parenthesis (parenthesis inside other parenthesis), we so go back to rule 1:
Do all calculations for operations inside of parenthesis.
(3 + (2 * 5 + (11) - 9)) * (6)
4 There are still parenthesis inside of other parenthesis, so calculate them in the same order of operations: Radicals/Exponents, Multiplication/Division, Addition/Subtraction (3 + (10 + 11 - 9)) * 6
5 We still have parenthesis, but we are slowly getting rid of them. Now that we only have addition and subtraction inside the inner parenthesis, this will be the last "inner parenthesis" calculation! (3 + (12)) * 6
6 Now we only have 1 group of parenthesis that require further calculations. (The parenthesis around 12 is just to show our previous calculation; there is no work to be done on "12" by itself.) (15) * 6
7 We are now done with parenthesis that need calculations! The once-large problem is now a simple multiplication problem! 15 * 6 = 90

Remeber: There are only the 4 rules you need to keep in mind when performing the order of operations. As long as you have parenthesis inside of parenthesis, perform each of the other rules to the inner-most parenthesis to calculate it's value, and work backwards to eliminate all the parenthesis to get a simple arithmetic problem!

Find another math lesson at the Math Lesson and Tutorial Center!

 

 

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