To determine the value of a quantity such as
1
2
+
5
8
⋅
2
15
1
2
+
5
8
⋅
2
15
size 12{ { {1} over {2} } + { {5} over {8} } cdot { {2} over {"15"} } } {}
where we have a combination of operations (more than one operation occurs), we must use the accepted order of operations.
- In the order (2), (3), (4) described below, perform all operations inside grouping symbols: ( ), [ ], ( ),
. Work from the innermost set to the outermost set.
- Perform exponential and root operations.
- Perform all multiplications and divisions moving left to right.
- Perform all additions and subtractions moving left to right.
Determine the value of each of the following quantities.
1
4
+
5
8
⋅
2
15
1
4
+
5
8
⋅
2
15
size 12{ { {1} over {4} } + { {5} over {8} } cdot { {2} over {"15"} } } {}
- (a) Multiply first.
1
4
+
5
1
8
4
⋅
2
1
15
3
=
1
4
+
1
⋅
1
4
⋅
3
=
1
4
+
1
12
1
4
+
5
1
8
4
⋅
2
1
15
3
=
1
4
+
1
⋅
1
4
⋅
3
=
1
4
+
1
12
size 12{ { {1} over {4} } + { { {5} cSup { size 8{1} } } over { {8} cSub { size 8{4} } } } cdot { { {2} cSup { size 8{1} } } over { {"15"} cSub { size 8{3} } } } = { {1} over {4} } + { {1 cdot 1} over {4 cdot 3} } = { {1} over {4} } + { {1} over {"12"} } } {}
- (b) Now perform this addition. Find the LCD.
4
=
2
2
12
=
2
2
⋅
3
The LCD
=
2
2
⋅
3
=
12
.
4
=
2
2
12
=
2
2
⋅
3
The LCD
=
2
2
⋅
3
=
12
.
1
4
+
1
12
=
1
⋅
3
12
+
1
12
=
3
12
+
1
12
=
3
+
1
12
=
4
12
=
1
3
1
4
+
1
12
=
1
⋅
3
12
+
1
12
=
3
12
+
1
12
=
3
+
1
12
=
4
12
=
1
3
Thus,
14+58⋅215=1314+58⋅215=13 size 12{ { {1} over {4} } + { {5} over {8} } cdot { {2} over {"15"} } = { {1} over {3} } } {}
3
5
+
9
44
5
9
−
1
4
3
5
+
9
44
5
9
−
1
4
size 12{ { {3} over {5} } + { {9} over {"44"} } left ( { {5} over {9} } - { {1} over {4} } right )} {}
- (a) Operate within the parentheses first,
59−1459−14 size 12{ left ( { {5} over {9} } - { {1} over {4} } right )} {}.
9
=
3
2
4
=
2
2
The LCD
=
2
2
⋅
3
2
=
4
⋅
9
=
36
.
9
=
3
2
4
=
2
2
The LCD
=
2
2
⋅
3
2
=
4
⋅
9
=
36
.
5
⋅
4
36
−
1
⋅
9
36
=
20
36
−
9
36
=
20
−
9
36
=
11
36
5
⋅
4
36
−
1
⋅
9
36
=
20
36
−
9
36
=
20
−
9
36
=
11
36
size 12{ { {5 cdot 4} over {"36"} } - { {1 cdot 9} over {"36"} } = { {"20"} over {"36"} } - { {9} over {"36"} } = { {"20" - 9} over {"36"} } = { {"11"} over {"36"} } } {}
Now we have
3
5
+
9
44
11
36
3
5
+
9
44
11
36
size 12{ { {3} over {5} } + { {9} over {"44"} } left ( { {"11"} over {"36"} } right )} {}
- (b) Perform the multiplication.
3
5
+
9
1
44
4
⋅
11
1
36
4
=
3
5
+
1
⋅
1
4
⋅
4
=
3
5
+
1
16
3
5
+
9
1
44
4
⋅
11
1
36
4
=
3
5
+
1
⋅
1
4
⋅
4
=
3
5
+
1
16
size 12{ { {3} over {5} } + { { {9} cSup { size 8{1} } } over { {"44"} cSub { size 8{4} } } } cdot { { {"11"} cSup { size 8{1} } } over { {"36"} cSub { size 8{4} } } } = { {3} over {5} } + { {1 cdot 1} over {4 cdot 4} } = { {3} over {5} } + { {1} over {"16"} } } {}
- (c) Now perform the addition. The LCD=80.
3
5
+
1
16
=
3
⋅
16
80
+
1
⋅
5
80
=
48
80
+
5
80
=
48
+
5
80
=
53
80
3
5
+
1
16
=
3
⋅
16
80
+
1
⋅
5
80
=
48
80
+
5
80
=
48
+
5
80
=
53
80
size 12{ { {3} over {5} } + { {1} over {"16"} } = { {3 cdot "16"} over {"80"} } + { {1 cdot 5} over {"80"} } = { {"48"} over {"80"} } + { {5} over {"80"} } = { {"48"+5} over {"80"} } = { {"53"} over {"80"} } } {}
Thus,
35+94459−14=538035+94459−14=5380 size 12{ { {3} over {5} } + { {9} over {"44"} } left ( { {5} over {9} } - { {1} over {4} } right )= { {"53"} over {"80"} } } {}
8
−
15
426
2
−
1
4
15
3
1
5
+
2
1
8
8
−
15
426
2
−
1
4
15
3
1
5
+
2
1
8
size 12{8 - { {"15"} over {"426"} } left (2 - 1 { {4} over {"15"} } right ) left (3 { {1} over {5} } +2 { {1} over {8} } right )} {}
- (a) Work within each set of parentheses individually.
2
−
1
4
15
=
2
1
⋅
15
+
4
15
=
2
−
19
15
=
30
15
−
19
15
=
30
−
19
15
=
11
15
3
1
5
+
2
1
8
=
3
⋅
5
+
1
5
+
2
⋅
8
+
1
8
=
16
5
+
17
8
LCD
=
40
=
16
⋅
8
40
+
17
⋅
5
40
=
128
40
+
85
40
=
128
+
85
40
=
213
40
2
−
1
4
15
=
2
1
⋅
15
+
4
15
=
2
−
19
15
=
30
15
−
19
15
=
30
−
19
15
=
11
15
3
1
5
+
2
1
8
=
3
⋅
5
+
1
5
+
2
⋅
8
+
1
8
=
16
5
+
17
8
LCD
=
40
=
16
⋅
8
40
+
17
⋅
5
40
=
128
40
+
85
40
=
128
+
85
40
=
213
40
Now we have
8
−
15
426
11
15
213
40
8
−
15
426
11
15
213
40
size 12{8 - { {"15"} over {"426"} } left ( { {"11"} over {"15"} } right ) left ( { {"213"} over {"40"} } right )} {}
- (b) Now multiply.
8
−
15
1
426
2
⋅
11
15
1
⋅
213
1
40
=
8
−
1
⋅
11
⋅
1
2
⋅
1
⋅
40
=
8
−
11
80
8
−
15
1
426
2
⋅
11
15
1
⋅
213
1
40
=
8
−
1
⋅
11
⋅
1
2
⋅
1
⋅
40
=
8
−
11
80
size 12{8 - { { {"15"} cSup { size 8{1} } } over { {"426"} cSub { size 8{2} } } } cdot { {"11"} over { {"15"} cSub { size 8{1} } } } cdot { { {"213"} cSup { size 8{1} } } over {"40"} } =8 - { {1 cdot "11" cdot 1} over {2 cdot 1 cdot "40"} } =8 - { {"11"} over {"80"} } } {}
- (c)
Now subtract.
8
−
11
80
=
80
⋅
8
80
−
11
80
=
640
80
−
11
80
=
640
−
11
80
=
629
80
or
7
69
80
8
−
11
80
=
80
⋅
8
80
−
11
80
=
640
80
−
11
80
=
640
−
11
80
=
629
80
or
7
69
80
size 12{8 - { {"11"} over {"80"} } = { {"80" cdot 8} over {"80"} } - { {"11"} over {"80"} } = { {"640"} over {"80"} } - { {"11"} over {"80"} } = { {"640" - "11"} over {"80"} } = { {"629"} over {"80"} } " or "7 { {"69"} over {"80"} } } {}
Thus,
8
-
15
426
2
-
1
4
15
3
1
5
+
2
1
8
=
7
69
80
8
-
15
426
2
-
1
4
15
3
1
5
+
2
1
8
=
7
69
80
3
4
2
⋅
8
9
−
5
12
3
4
2
⋅
8
9
−
5
12
size 12{ left ( { {3} over {4} } right ) rSup { size 8{2} } cdot { {8} over {9} } - { {5} over {"12"} } } {}
- (a) Square
3434 size 12{ { {3} over {4} } } {}.
3
4
2
=
3
4
⋅
3
4
=
3
⋅
3
4
⋅
4
=
9
16
3
4
2
=
3
4
⋅
3
4
=
3
⋅
3
4
⋅
4
=
9
16
size 12{ left ( { {3} over {4} } right ) rSup { size 8{2} } = { {3} over {4} } cdot { {3} over {4} } = { {3 cdot 3} over {4 cdot 4} } = { {9} over {"16"} } } {}
Now we have
9
16
⋅
8
9
−
5
12
9
16
⋅
8
9
−
5
12
size 12{ { {9} over {"16"} } cdot { {8} over {9} } - { {5} over {"12"} } } {}
- (b) Perform the multiplication.
9
1
16
2
⋅
8
1
9
1
−
5
12
=
1
⋅
1
2
⋅
1
−
5
12
=
1
2
−
5
12
9
1
16
2
⋅
8
1
9
1
−
5
12
=
1
⋅
1
2
⋅
1
−
5
12
=
1
2
−
5
12
size 12{ { { {9} cSup { size 8{1} } } over { {"16"} cSub { size 8{2} } } } cdot { { {8} cSup { size 8{1} } } over { {9} cSub { size 8{1} } } } - { {5} over {"12"} } = { {1 cdot 1} over {2 cdot 1} } - { {5} over {"12"} } = { {1} over {2} } - { {5} over {"12"} } } {}
- (c) Now perform the subtraction.
1
2
−
5
12
=
6
12
−
5
12
=
6
−
5
12
=
1
12
1
2
−
5
12
=
6
12
−
5
12
=
6
−
5
12
=
1
12
size 12{ { {1} over {2} } - { {5} over {"12"} } = { {6} over {"12"} } - { {5} over {"12"} } = { {6 - 5} over {"12"} } = { {1} over {"12"} } } {}
Thus,
432⋅89−512=112432⋅89−512=112 size 12{ left ( { {4} over {3} } right ) rSup { size 8{2} } cdot { {8} over {9} } - { {5} over {"12"} } = { {1} over {"12"} } } {}
2
7
8
+
25
36
÷
2
1
2
−
1
1
3
2
7
8
+
25
36
÷
2
1
2
−
1
1
3
size 12{2 { {7} over {8} } + sqrt { { {"25"} over {"36"} } } div left (2 { {1} over {2} } - 1 { {1} over {3} } right )} {}
- (a) Begin by operating inside the parentheses.
2
1
2
−
1
1
3
=
2
⋅
2
+
1
2
−
1
⋅
3
+
1
3
=
5
2
−
4
3
=
15
6
−
8
6
=
15
−
8
6
=
7
6
2
1
2
−
1
1
3
=
2
⋅
2
+
1
2
−
1
⋅
3
+
1
3
=
5
2
−
4
3
=
15
6
−
8
6
=
15
−
8
6
=
7
6
- (b) Now simplify the square root.
25
36
=
5
6
since
5
6
2
=
25
36
25
36
=
5
6
since
5
6
2
=
25
36
size 12{ sqrt { { {"25"} over {"36"} } } = { {5} over {6} } left ("since " left ( { {5} over {6} } right ) rSup { size 8{2} } = { {"25"} over {"36"} } right )} {}
Now we have
2
7
8
+
5
6
÷
7
6
2
7
8
+
5
6
÷
7
6
size 12{2 { {7} over {8} } + { {5} over {6} } div { {7} over {6} } } {}
- (c) Perform the division.
2
7
8
+
5
6
1
⋅
6
1
7
=
2
7
8
+
5
⋅
1
1
⋅
7
=
2
7
8
+
5
7
2
7
8
+
5
6
1
⋅
6
1
7
=
2
7
8
+
5
⋅
1
1
⋅
7
=
2
7
8
+
5
7
size 12{2 { {7} over {8} } + { {5} over { {6} cSub { size 8{1} } } } cdot { { {6} cSup { size 8{1} } } over {7} } =2 { {7} over {8} } + { {5 cdot 1} over {1 cdot 7} } =2 { {7} over {8} } + { {5} over {7} } } {}
- (d) Now perform the addition.
2
7
8
+
5
7
=
2
⋅
8
+
7
8
+
5
7
=
23
8
+
5
7
LCD
=
56
.
=
23
⋅
7
56
+
5
⋅
8
56
=
161
56
+
40
56
=
161
+
40
56
=
201
56
or
3
33
56
2
7
8
+
5
7
=
2
⋅
8
+
7
8
+
5
7
=
23
8
+
5
7
LCD
=
56
.
=
23
⋅
7
56
+
5
⋅
8
56
=
161
56
+
40
56
=
161
+
40
56
=
201
56
or
3
33
56
Thus,
278+2536÷212−113=33356278+2536÷212−113=33356 size 12{2 { {7} over {8} } + sqrt { { {"25"} over {"36"} } } div left (2 { {1} over {2} } - 1 { {1} over {3} } right )=3 { {"33"} over {"56"} } } {}
Find the value of each of the following quantities.
516⋅110−132516⋅110−132 size 12{ { {5} over {"16"} } cdot { {1} over {"10"} } - { {1} over {"32"} } } {}
67⋅2140÷910+51367⋅2140÷910+513 size 12{ { {6} over {7} } cdot { {"21"} over {"40"} } div { {9} over {"10"} } +5 { {1} over {3} } } {}
356356 size 12{ { {"35"} over {6} } } {} or
556556 size 12{5 { {5} over {"6"} } } {}
8710−2412−3238710−2412−323 size 12{8 { {7} over {"10"} } - 2 left (4 { {1} over {2} } - 3 { {2} over {3} } right )} {}
2113021130 size 12{ { {"211"} over {"30"} } } {} or
71307130 size 12{7 { {1} over {"30"} } } {}
1718−583014−3321−13291718−583014−3321−1329 size 12{ { {"17"} over {"18"} } - { {"58"} over {"30"} } left ( { {1} over {4} } - { {3} over {"32"} } right ) left (1 - { {"13"} over {"29"} } right )} {}
7979 size 12{ { {7} over {9} } } {}
110+112÷145−1625110+112÷145−1625 size 12{ left ( { {1} over {"10"} } +1 { {1} over {2} } right ) div left (1 { {4} over {5} } - 1 { {6} over {"25"} } right )} {}
267267 size 12{2 { {6} over {7} } } {}
23−38⋅49716⋅113+11423−38⋅49716⋅113+114 size 12{ { { { {2} over {3} } - { {3} over {8} } cdot { {4} over {9} } } over { { {7} over {"16"} } cdot 1 { {1} over {3} } +1 { {1} over {4} } } } } {}
311311 size 12{ { {3} over {"11"} } } {}
382+34⋅18382+34⋅18 size 12{ left ( { {3} over {8} } right ) rSup { size 8{2} } + { {3} over {4} } cdot { {1} over {8} } } {}
15641564 size 12{ { {"15"} over {"64"} } } {}
23⋅214−42523⋅214−425 size 12{ { {2} over {3} } cdot 2 { {1} over {4} } - sqrt { { {4} over {"25"} } } } {}
11101110 size 12{ { {"11"} over {"10"} } } {}
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