Question 439478
first something about the {{{ORDER}}} OF OPERATIONS RULES

When performing more than one operation on an algebraic expression, work out the operations and signs in the following order:

    {{{First}}} calculate powers and roots.

    {{{Then}}} perform all multiplication and division.

    Finally, {{{finish}}} with addition and subtraction.

{{{Order}}} of operations are a {{{set}}} of {{{rules}}} that mathematicians have {{{agreed}}} to {{{follow}}} to avoid mass {{{CONFUSION}}} when {{{simplifying}}} mathematical expressions or equations.


easier way to remember the ORDER OF OPERATIONS RULES

For those of you that remember best with acronyms:

{{{Please}}}{{{ Excuse }}}{{{My}}}{{{ Dear}}}{{{ Aunt}}}{{{ Sally}}} ({{{PEMDAS}}})

    Please ...=>...Parentheses

    Escuse...=>...Exponents

    My ...=>...Multiplication

    Dear ...=>...Division

    Aunt ...=>...Addition

    Sally ...=>...Subtraction

now, about the {{{RE-ORDER}}} OF OPERATIONS RULES  

the {{{Associative}}} Property, the {{{commutative}}} property, and the {{{distributive}}} property-three basic properties of numbers- allow you to {{{move}}}{{{ stuff}}}{{{ around}}}, {{{regroup}}}, all {{{without}}} affecting the result


the Associative Property-Use the associative property to change the grouping in an algebraic expression to make the work tidier or more convenient.


The commutative property makes working with algebraic expressions easier. The commutative property changes the order of some numbers in an operation to make the work tidier or more convenient — all without affecting the result.

Addition: {{{a + b = b + a}}}

Example: {{{4 + 5 = 9}}} and {{{5 + 4 = 9}}}, so {{{4 + 5 = 5 + 4}}}

{{{Reordering}}} the numbers {{{doesn't}}} affect the result.


Multiplication:{{{a * b = b * a}}}

Subtraction: {{{a – b }}} is not equal to {{{ b – a }}}(except in a few special cases)

Example: (–5) – (+2) = (–7) and (+2) – (–5) = +7, so (–5) – (+2) is not equal to  (+2) – (–5)

Here, you see how {{{subtraction}}} {{{doesn't}}} follow the commutative property.

Exception: If {{{a}}} and {{{b}}} are the same number, then the subtraction appears to be commutative because switching the order doesn’t change the answer.

Example: 2 –2 = 0 and –2 + 2 = 0, so 2 –2 = –2 + 2



Division: {{{a :b not= b χ a}}} (except in a few special cases)

Example: {{{(-6) : (+1) = -6}}} and {{{(+1) : (-6) = -1/6}}}, so {{{(-6) χ (+1)}}} is not equal to {{{(+1) / -6)}}}

Division also {{{doesn't}}} follow the commutative property.

Exception: If a and b are opposites, then you get –1 no matter which order you divide them in.

Example: 2 : –2 = –1 and –2:2 = –1, so 2:–2 = –2 :2