The order in which mathematical operations are performed must be the same in every case.  Otherwise we wouldn't get the same results when we simplified expressions involving parentheses and exponents and such.

Here is the order of operations:

     [1] Parentheses.  Go through the expression, from left to right, and do nothing except simplify, where possible, any expressions that are encased in parentheses.  This means, then, that the innermost parentheses in any expression is what gets simplified first.

                        Example:   3 + (5-9)  becomes 3 + (-4), which becomes -1.

    [2] Exponents:  The second time through the expression, do nothing except carry out any exponentiation that is indicated.

                       Example:   5 + 3(2x+4) - 8².  becomes 5 + 3(2x + 4) - 64

                       You will notice some parentheses in the above example.  The reason the expression inside them has not first been simplified is because it cannot be:  The first term inside them is an "x" term, and the second term is a constant term.  Not being like terms, they cannot be combined.

    [3] Multiplication/Division.  On your third time through the expression, do nothing except carry out any multiplication or division as it       occurs (from left to right).

                      Example:  5 + 3(2x + 4) - 64 = 5 + 6x + 12 - 64.

    [4] Addition/Subtraction.   This operation can be thought of as "combining like terms," which is the last thing to be done in the order of operations.  In the example above, the only like terms are the constants, and so we get:

                     Example:  5 + 6x + 12 - 64 = 6x + 5 +12 - 64 = 6x + 17 - 64 = 6x - 47

Now, remember:  when you simplify an expression encased in parentheses, you must use this order on the expression that is inside the parentheses.

Here is a sample problem that tests your knowledge of the order of operations:

           Simplify:     3 + 4[(6-9)-8] - 4² + 2[3(2-5) + 9]

    [1]    3 + 4[(-3)-8] - 4² + 2[3(-3) + 9] = 3 + 4(-11) - 4² + 2[-9 + 9] = 3 + 4(-11) - 4² + 2[0]

                                       Now the parentheses have been resolved.

    [2]   3 + 4(-11) - 4² + 2[0] = 3 + 4(-11) - 16 + 2(0)

                                      The exponentiation has been carried out.

    [3]   3 - 44 - 16 + 0

                                      The multiplication and/or division has been done.            

    [4]  3 - 60 = -57
                                      Lastly, the addition/subtraction, or combining like terms, gives us the simplified value 
                                      of the expression  we started with.

The phrase that has been used to remember the order of operations is:  Please Excuse My Dear Aunt Sally.