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.
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(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, I'll simplify inside the parentheses, and then inside 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 − (1)(2x) − (1)(+3))]
14x + 5[6 − 2x − 3]
14x + 5[6 − 3 − 2x]
14x + 5[3 − 2x]
14x + 5(3) + 5(−2x)
14x + 15 − 10x
14x − 10x + 15
4x + 15
It isn't required that you rearrange the terms to group "like" terms together, but it's generally a good idea to do so, certainly when you're just starting out. But, by grouping appropriately, it's a lot harder to lose terms when you're adding things up.
I need to remember to simplify at each step, combining like terms when and where I can. I'll start by inserting the "understood" 1's in front of the subtracted grouping symbols:
−{2x − [3 − (4 − 3x)] + 6x}
−1{2x − 1[3 − 1(4 − 3x)] + 6x}
−1{2x − 1[3 − 1(4) − 1(−3x)] + 6x}
−1{2x − 1[3 − 4 + 3x] + 6x}
−1{2x − 1[−1 + 3x] + 6x}
−1{2x − 1(−1) −1(+3x) + 6x}
−1{2x + 1 − 3x + 6x}
−1{2x + 6x − 3x + 1}
−1{5x + 1}
−1(5x) − 1(+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 fractional terms, I first have to simplify each term.
As it happens, each of the fractions above simplified to whole numbers, so I didn't have to muck about with common denominators. You will not usually be so lucky.
To do this simplification, I have to work the top and 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 to the next page.)
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