Sometimes there are no grouping symbols indicating which operations to perform first. For example, suppose we wish to find the value of
3 + 5⋅ 23 + 5⋅ 2 size 12{"3 "+" 5" cdot " 2"} {}. We could do either of two things:
Add 3 and 5, then multiply this sum by 2.
3+5⋅2=8⋅2=163+5⋅2=8⋅2=16alignl { stack {
size 12{`3+5 cdot 2} {} #
size 12{`=8 cdot 2} {} #
size 12{`="16"} {}
} } {}
Multiply 5 and 2, then add 3 to this product.
3+5⋅2=3+10=133+5⋅2=3+10=13alignl { stack {
size 12{`3+5 cdot 2} {} #
size 12{`=3+"10"} {} #
size 12{`="13"} {}
} } {}
We now have two values for the same expression.
We need a set of rules to guide anyone to one unique value for this kind of expression. Some of these rules are based on convention, while other are forced on up by mathematical logic.
The universally agreed-upon accepted order of operations for evaluating a mathematical expression is as follows:
1. Parentheses (grouping symbols) from the inside out.
By parentheses we mean anything that acts as a grouping symbol, including anything inside symbols such as [ ], { }, | |, and
size 12{` sqrt {`} } {}. Any expression in the numerator or denominator of a fraction or in an exponent is also considered grouped, and should be simplified before carrying out further operations.
If there are nested parentheses (parentheses inside parentheses), you work from the innermost parentheses outward.
2. Exponents and other special functions, such as log, sin, cos etc.
3. Multiplications and divisions, from left to right.
4. Additions and subtractions, from left to right.
For example, given: 3 + 15 ÷ 3 + 5 × 22+3
The exponent is an implied grouping, so the 2+3 must be evaluated first:
= 3 + 15 ÷ 3 + 5 × 25
Now the exponent is carried out:
= 3 +15 ÷ 3 + 5 × 32
Then the multiplication and division, left to right using 15 ÷ 3 = 5 and 5 × 32 = 160:
= 3 + 5 + 160
Finally, the addition, left to right:
= 168
Determine the value of each of the following.
21+3⋅1221+3⋅12 size 12{"21"+3 cdot "12"} {}.
Multiply first:
=
21+3621+36 size 12{"21"+"36"} {}
Add.
= 57
(15−8)+5(6+4)(15−8)+5(6+4) size 12{ \( "15" - 8 \) +5 \( 6+4 \) } {}.
Simplify inside parentheses first.
=
7+5⋅107+5⋅10 size 12{7+5 cdot "10"} {}
Multiply.
=
7+507+50 size 12{7+"50"} {}
Add.
= 57
63−(4+6⋅3)+76−463−(4+6⋅3)+76−4 size 12{"63" - \( 4+6 cdot 3 \) +"76" - 4} {}.
Simplify first within the parentheses by multiplying, then adding:
=
63−(4+18)+76−463−(4+18)+76−4 size 12{"63" - \( 4+"18" \) +"76" - 4} {}
=
63−22+76−463−22+76−4 size 12{"63" - "22"+"76" - 4} {}
Now perform the additions and subtractions, moving left to right:
=
41+76−441+76−4 size 12{"41"+"76" - 4} {}
=
117−4117−4 size 12{"117" - 4} {}
= 113.
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size 12{7 cdot 6 - 4 rSup { size 8{2} } +1 rSup { size 8{5} } } {}
Evaluate the exponential forms, moving from left to right:
=
7⋅6−16+17⋅6−16+1 size 12{7 cdot 6 - "16"+1} {}
Multiply 7 · 6:
=
42−16+142−16+1 size 12{"42" - "16"+1} {}
Subtract 16 from 42:
= 26 + 1
Add 26 and 1:
= 27.
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size 12{6 cdot \( 3 rSup { size 8{2} } +2 rSup { size 8{2} } \) +4 rSup { size 8{2} } } {}
Evaluate the exponential forms in the parentheses:
=
6⋅(9+4)+426⋅(9+4)+42 size 12{6 cdot \( 9+4 \) +4 rSup { size 8{2} } } {}
Add 9 and 4 in the parentheses:
=
6⋅(13)+426⋅(13)+42 size 12{6 cdot \( "13" \) +4 rSup { size 8{2} } } {}
Evaluate the exponential form
4242 size 12{4 rSup { size 8{2} } } {}:
=
6⋅(13)+166⋅(13)+16 size 12{6 cdot \( "13" \) +"16"} {}
Multiply 6 and 13:
=
78+1678+16 size 12{"78"+"16"} {}
Add 78 and 16:
= 94
62+2242+6⋅22+13+82102−19⋅562+2242+6⋅22+13+82102−19⋅5 size 12{ { {6 rSup { size 8{2} } +2 rSup { size 8{2} } } over {4 rSup { size 8{2} } +6 cdot 2 rSup { size 8{2} } } } + { {1 rSup { size 8{3} } +8 rSup { size 8{2} } } over {"10" rSup { size 8{2} } - "19" cdot 5} } } {}.
=
36+416+6⋅4+1+64100−19⋅536+416+6⋅4+1+64100−19⋅5 size 12{ { {"36"+4} over {"16"+6 cdot 4} } + { {1+"64"} over {"100" - "19" cdot 5} } } {}
=
36+416+24+1+64100−9536+416+24+1+64100−95 size 12{ { {"36"+4} over {"16"+"24"} } + { {1+"64"} over {"100" - "95"} } } {}
=
4040+6554040+655 size 12{ { {"40"} over {"40"} } + { {"65"} over {5} } } {}
= 1+13
= 14
Recall that the bar is a grouping symbol. The fraction
62+2242+6⋅2262+2242+6⋅22 size 12{ { {6 rSup { size 8{2} } +2 rSup { size 8{2} } } over {4 rSup { size 8{2} } +6 cdot 2 rSup { size 8{2} } } } } {} is equivalent to
62+22÷42+6⋅2262+22÷42+6⋅22 size 12{ left (6 rSup { size 8{2} } +2 rSup { size 8{2} } right ) div left (4 rSup { size 8{2} } +6 cdot 2 rSup { size 8{2} } right )} {}
Determine the value of the following:
8(10) + 4(2 + 3) – (20 + 3 · 15 + 40 – 5)
(8 + 9 · 3) ÷ 7 + 5 · (8 ÷ 4 + 7 + 3 · 5)
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size 12{ { {3 rSup { size 8{3} } +2 rSup { size 8{3} } } over {6 rSup { size 8{2} } - "29"} } +5 left ( { {8 rSup { size 8{2} } +2 rSup { size 8{4} } } over {7 rSup { size 8{2} } - 3 rSup { size 8{2} } } } right ) div { {8 cdot 3+1 rSup { size 8{8} } } over {2 rSup { size 8{3} } - 3} } } {}
