Algebraic Techniques - Part 1a: The Order of Operations

Algebraic Techniques - Part 1a: The Order of Operations
An overview of how to evaluate mathematical expressions using the rules of the order of operations. To learn more about this topic buy our Numerical Expressions, Grouping Symbols & The Order of Operations Workbook: www.fatfreecar...
If there is more than one operation to be performed in a mathematical expression, there has to be universal agreement about which operations should be performed first, otherwise, depending on how a person chooses to evaluate an expression, there could be completely different results. For example, if we have this expression, should we add the 5 and 3 first to get 8, then multiply by the 2 to get 16? Or, should we multiply the 3 and 2 first, to get 6, and then add 5, to get 11?
There are five basic rules that mathematicians around the world have agreed to follow in evaluating expressions:
Do operations inside of grouping symbols such as parentheses first. Given this expression, we add the 3 and 5 first, to get 8, and then we multiply by 2 to get 16. If we have 2 plus 3 quantity cubed, we should add 2 and 3 first, to get 5, then cube the 5, to get 125.
If grouping symbols are nested, the operations inside of the innermost grouping symbols should be done first. If we want to evaluate the single-layer nested expression that we saw earlier, first we simplify 3 minus 9, to get negative 6. Multiply negative 2 by negative 6 to get positive 12. Then add the 5 and 12 to get 17, but since there is a negative sign outside of the square bracket, it is actually negative 17. Adding 4 and negative 17, we get negative 13.
If we have this expression, also with a single layer of nesting, first, subtract 1 from 9 to get 8. Multiply negative 2 and 8, to get negative 16. Then add 5 and negative 16, to get negative 11. Finally, multiply 3 and negative 11, to get negative 33.
Do exponentiations or powers and roots in any order. If we have 4 plus 3 squared, to get 4 plus 9, then add, to get 13. Given 3 times 2 cubed, cube 2, to get 3 times 8 and finally 24.
Stacked exponents should be evaluated from the uppermost down. is equal to because is equal to 8 and so is equal to 256.
Do multiplications and divisions from left to right. The first expression that we looked at is equal to 11, since we should multiply 3 and 2 first, to get 6, then add 5. Given 14 plus 16 divided by 4, divide 16 by 4 first, to get 14 plus 4 and a sum of 18.
Do additions and subtractions from left to right. Given this expression, square the 4 and 3 first, then do the division and multiplication and then add and subtract from left to right, to get negative 1.
Appearance at a higher level on the list indicates that the operation has a higher precedence in evaluations. Multiplication and division have equal precedence and so do addition and subtraction, but both have to be done at a specific point in the evaluation, depending on where they appear in the expression. When logarithms appear in expressions, later in the Algebra curriculum, they have the same precedence as exponentiation and roots.
Various mnemonic devices have been used to remember the Order of Operations. Of the most commonly used, one is the acronym P-E-M-D-A-S or ‘PEMDAS’, which is often expanded to Please-Excuse-My-Dear-Aunt-Sally, but in either case, the letters represent: Parentheses, Exponents, Multiplication, Division, Addition and Subtraction.
These rules apply to the purely numerical expressions of arithmetic as well as the symbolic and numerical expressions of every other branch of mathematics.
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