Overview: A Set of Rules
The order of operations refers to a set of rules that are commonly agreed upon in mathematics, so that the answer to a problem is always the same when the correct order is followed. If numbers are combined in a different order, then different answers will result. When numbers are replaced by variables, as in algebra, the problem becomes compounded.
Parentheses and Fractions are First
Expressions that are grouped within parentheses are solved first. Usually, they are grouped within parentheses or brackets because grouping the numbers differently would change the meaning. For example, suppose the number sentence is 3[15-(5+3)]. To solve it, the innermost parenthesis (5+3) equals 8, 15-8 = 7 and 3 X 7 =21. If the number sentence read 3[(15-5)+3] the innermost parenthesis would be 10 , and the problem would read 3 (13) which would equal 39. Parentheses, brackets, and fraction bars are signals that numbers are to be grouped together.
Multiplication and Division Are Next
Multiplication and its inverse, division, are next. They go in order from left to right. Exponents are a form of multiplication, and square roots are a form of division. For example if the problem is 2 X 18 + 4 X 35, it is solved as if it were read (2 X 18) + (4 X 35), or 36 + 140 = 176. Because of the rules of order of operations, the parentheses are not necessary.
Finally, Addition and Subtraction
Last, addition and its inverse, subtraction, are done. If there are no signals such as parentheses that numbers are to be grouped together, those operations are performed last. That is another way to ensure that the problem has one unique answer. Students can always read the problem in order to evaluate what is in parentheses or grouped in fractions, what numbers need to be multiplied or divided, and then what needs to be added or subtracted to solve for a unique answer.
Scientific Calculators Follow the Rules
Scientific calculators have the rules programmed into them. In order to use a calculator as a shortcut to solve a mathematical problem, it is very important to remember to enter the parentheses to work the problem correctly. The parentheses are a clue that numbers are being combined in a different order, and are an essential part of the number sentence. If there are no parentheses in the number sentence, the numbers and operations are entered in order from left to right. The scientific calculator will use the rules of the order of operations to solve for the correct answer.
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