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The Order of Operations: More Examples (page 2 of 3)


  • Simplify 4 3[4 2(6 3)] 2.
  • I will simplify from the inside out: first the parentheses, then the square brackets, being careful to remember that the "minus" sign on the 3 in front of the brackets goes with the 3. Only once the grouping parts are done will I do the division, followed by adding in the 4.

      4 3[4 2(6 3)] 2
          = 4 3[4 2(3)] 2

          = 4 3[4 6] 2

          = 4 3[2] 2

          = 4 + 6 2

          = 4 + 3

          =
      7

Remember that, in leiu of grouping symbols telling you otherwise, the division comes before the addition, which is why this expression simplified, in the end, down to "4 + 3", and not "10 2".

 

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  • Simplify 16 3(8 3)2 5.
  • I must remember to simplify inside the parentheses before I square, because (8 3)2 is not the same as 82 32.

      16 3(8 3)2 5
          = 16 3(5)2 5

          = 16 3(25) 5

          = 16 75 5

          = 16 15

          =
      1


If you have learned about variables and combining "like" terms, you may also see exercises such as this:   Copyright Elizabeth Stapel 2000-2011 All Rights Reserved

  • Simplify 14x + 5[6 (2x + 3)].
  • If I have trouble taking a subtraction through a parentheses, I can turn it into multiplying a negative 1 through the parentheses (note the highlighted red "1" below):

      14x + 5[6 (2x + 3)]
          = 14x + 5[6
      1(2x + 3)]
          = 14x + 5[6 2x 3]

          = 14x + 5[3 2x]

          = 14x + 15 10x

          =
      4x + 15

  • Simplify {2x [3 (4 3x)] + 6x}.
  • I need to remember to simplify at each step, combining like terms when and where I can:

      {2x [3 (4 3x)] + 6x}
          =
      1{2x 1[3 1(4 3x)] + 6x}
          =
      1{2x 1[3 4 + 3x] + 6x}
          =
      1{2x 1[ 1 + 3x] + 6x}
          =
      1{2x + 1 3x + 6x}
          =
      1{2x + 6x 3x + 1}
          =
      1{5x + 1}
          =
      5x 1

(For more examples of this sort, review Simplifying with Parentheses.)


This next example displays an issue that almost never arises but, when it does, there seems to be no end to the arguing.

  • Simplify 16 2[8 3(4 2)] + 1.
    • 16 2[8 3(4 2)] + 1
          = 16 2[8 3(2)] + 1

          = 16 2[8 6] + 1

          = 16 2[2] + 1  
      (**)
          = 16 4 + 1
          = 4 + 1

          =
      5

The confusing part in the above calculation is how "16 divided by 2[2] + 1" (in the line marked with the double-star) becomes "16 divided by 4 + 1", instead of "8 times by 2 + 1". That's because, even though multiplication and division are at the same level (so the left-to-right rule should apply), parentheses outrank division, so the first 2 goes with the [2], rather than with the "16 divided by". That is, multiplication that is indicated by placement against parentheses (or brackets, etc) is "stronger" than "regular" multiplication. Typesetting the entire problem in a graphing calculator verifies this hierarchy:

    calculator screen-shot: 16 / 2(8) = 1, but 16 / 2 * 8 = 64

Note that different software will process this differently; even different models of Texas Instruments graphing calculators will process this differently. In cases of ambiguity, be very careful of your parentheses, and make your meaning clear. The general consensus among math people is that "multiplication by juxtaposition" (that is, multiplying by just putting things next to each other, rather than using the "" sign) indicates that the juxtaposed values must be multiplied together before processing other operations. But not all software is programmed this way, and sometimes teachers view things differently. If in doubt, ask!

(And please do not send me an e-mail either asking for or else proffering a definitive verdict on this issue. As far as I know, there is no such final verdict. And telling me to do this your way will not solve the issue!)

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Cite this article as:

Stapel, Elizabeth. "The Order of Operations: More Examples." Purplemath. Available from
    http://www.purplemath.com/modules/orderops2.htm. Accessed
 

 



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