The Order of Operations: Examples
Purplemath
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.)
- Simplify 4 − 3[4 −2(6 − 3)] ÷ 2
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.)
- 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:
-
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 − (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.
- Simplify −{2x − [3 − (4 − 3x)] + 6x}
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.)
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.
- Simplify
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.
- Simplify
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.)
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