Most of the issues with simplifying using the order of operations stem from nested parentheses, exponents, and "minus" signs. So, in the examples that follow, I'll be demonstrating how to work with these sorts of expressions.
(Links are provided for additional review of working with negatives, grouping symbols, and powers.)
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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".
(If you're not feeling comfortable with all of those "minus" signs, review Negatives.)
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I must remember to simplify inside the parentheses before I square, because (8 – 3)^{2} is not the same as 8^{2} – 3^{2}.
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:
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
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.)
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Expressions containing fractional forms can cause confusion, too. But, as long as you work the numerator (that is, the top) and the denominator (that is, the bottom) separately, until they're completely simplified first, and only then combine (or reduce), if possible, then you should be fine. If a fractional form is added to, or subtracted from, another term, fractional or otherwise, make sure you've completely simplified and reduced the fractional form before you try to do the addition or subtraction.
Before I can add the two terms, I must simplify.
This works just like the previous examples. I just have to work a "top" and a "bottom" separately, until I get a fraction that I can (possibly) reduce.
(For examples with loads of exponents, review Simplifying with Exponents.)
You can use the Mathway widget below to practice simplifying using the order of operations. Try the entered exercise, or type in your own exercise. Then click the button and select "Simplify" or "Evaluate" from the pop-up box to compare your answer to Mathway's. (Or skip the widget and continue with the lesson.)
(Click "Tap to view steps" to be taken directly to the Mathway site for a paid upgrade.)
This next example displays an issue that almost never arises but, when it does, there seems to be no end to the arguing. (It's become annoying popular to post these to Facebook.)
I simplify in the usual way:
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 seem somehow to outrank division, so the first 2 in the starred line is often regarded as going with the [2] that follows it, rather than with the "16 divided by" that precedes it. That is, the multiplication that is indicated by placement against parentheses (or brackets, etc) is often regarded (by science-y folks) as being "stronger" somehow than "regular" multiplication which is indicated by a symbol of some sort, such as "×".
Typesetting the entire problem in a graphing calculator verifies the existence of this hierarchy, at least in some software:
Note that different software packages will process this expression differently; even different models of Texas Instruments graphing calculators will process this expression differently. 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, when typing things out sideways, be very careful of your parentheses, and make your meaning clear, so as to avoid precisely this ambiguity.
(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. Telling me to do things your way will not solve the issue!) (For an example of the sort of e-mails I get on this, continue to the next page, which also contains more fractional-form examples.)
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