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1.02 Order
of Operations from Basic Algebra: One Step at a Time © 2002 P. 7-12 Dr. Robert J. Rapalje Seminole State College of Florida
ANSWERS TO ALL EXERCISES ARE INCLUDED AT THE END OF THIS PAGE
In
the first section, different number systems were defined, and symbols
relating to addition (subtraction) and multiplication (division) of
numbers within those systems were identified.
In order to execute these basic operations it is necessary to
establish a few traditions (agreements!) as to how these operations will
be carried out. The signs of
the numbers must also be addressed. In
this section it is preferred that you use the calculator “in your
head.” In the next section, you will be encouraged to use the
calculator “in your hand.” If
you do not already have one, be looking for a good sale on inexpensive
calculators! ORDER OF OPERATIONS In
your head, perform the following calculations: 1.
4 + 6 ·
2 = _______
2.
16 ÷ 8 ·
2 = _______ 3. 5 ·
2
2
= _______
4. (3 + 4)
2 = _______ Solutions: 1.
If
addition is performed first, the answer is 20.
If
multiplication is performed first, the answer is 16. 2.
If division is performed
first, the answer is 4.
If
multiplication is performed first, the answer is 1. 3.
If multiplication is
performed first, the answer is 100.
If the power is
performed first, the answer is 20. 4.
If
you add first, the answer is 49.
If you square the 3 and then square the 4 (which the problem does not
say to do!),
then the answer
would be 25. (SEE NEXT PARAGRAPH!! ) A
mathematics system must have unique solutions to basic operations such as
these. In order to perform
these operations and always
arrive at the same conclusion, we must agree to certain traditions which
are called the order of operations
agreement. This agreement
is generally accepted across the world and in most arenas of life.
If you obey these traditions, you will obtain consistent results in
agreement with everyone else.
Now,
we can say of the examples on the previous page, the “correct answers”
are: 1. 16;
2.
4;
3. 20;
4. 49. EXERCISES:
Simplify each of the following
according to the order of operations agreement. 1.
2 + 8 ·
4
2.
20 ÷ 4 ·
5
3.
2 ·
5 2 =
__________
= __________
=
__________ =
__________
= __________
= __________ 4.
32 ÷ 4 ·
8
5.
100 ÷ 5 2
6.
(2 ·
5) 2
=
__________
= __________
= __________ =
__________
= __________
= __________ 7.
12 + 8 ÷ 2
8.
12 ÷ (3 + 3)
9.
5 + 9 · 2
= __________
= __________
= __________
= __________
= __________
=
__________ 10.
(100 ÷ 5) 2
11.
12 ÷ 3 + 3
12.
(5 + 9) · 2
= __________
= __________
= __________
= __________
= __________
= __________ 13.
20 - 5 ·
0 14.
12 + 3 ÷ 3
15.
12 ÷ (3 ÷ 3)
= __________
=
__________
= __________
= __________
= __________
= ___________ 16.
5 + 5 2
17.
(12 + 3) ÷ 3
18.
18 + 2 · 0
= __________
= __________
= __________
= __________
= __________
=
__________ 19.
12 - 8 ÷ 2
20.
12 ÷ (3
·
2)
21.
12 ÷ 3 ·
2
= __________
= __________
= __________
= __________
= __________
= __________ 22.
(12 - 8) ÷ 2
23.
(12 - 8)2 ÷ 2
24.
20 - (8 ÷ 2) 2
= __________
=
__________
= __________
= __________
= __________
= __________ 25. (12
- 8 ÷ 2) 2
26.
(12 - 8) ÷ 22
27.
20 + (8 ÷ 2) 2
= __________
= __________
= __________
= __________
= __________
=
__________ 28.
5 ·
5 2
29.
( 5 ·
5) 2
30.
18 ÷ 2 ·
3
= __________
= __________
= __________
= __________
= __________
= __________ 31.
8 + 3 ·
2 2
32.
5 ·
3 2 + 11 - 2 2
= _______________
=
_______________
= _______________
= _______________
= _______________
= _______________
33.
24 - 3 ·
4 + 6 ÷ 2
34.
12 + 18 ÷ 3 2
+ 6
= _______________
= _______________
= _______________
= _______________ 35.
12 + (18 ÷ 3)2 + 6
36.
6 + 6 2 ÷ 3 · 2
= _______________
= _______________
= _______________
= _______________
= _______________
= _______________
= _______________ 37.
(16 + 22
) ÷ 2 ·
2
38.
8 + 2 ·
3 2 - 5
= _______________
= _______________ = _______________
= _______________
= _______________
= _______________
= _______________
= _______________ 39.
52
+ 42 ÷ 22 + 32
40.
(16 + 22 ) ÷ (2 + 2)
= _______________
= _______________
= _______________
= _______________
= _______________
= _______________
= _______________
= _______________ EXTRA
CHALLENGE: 41.
3·42
+ 7 - 2·32
÷ 6 + 3·52
42.
24 - 12 ÷ 2·3
+ 6·23
43.
(5 ·
2)2 - 20 ÷ 2 · (8 - 3) + 10 - 3 + 7
44. 35 -
20 ÷ 5 + 72 · 2 - 6 · 3 + 9 + 10 ÷ 2
45.
47.
ANSWERS
1.02 p. 8 - 12:
1. 34;
2. 25; 3. 50; 4.
64; 5. 4; 6. 100; 7.
16; 8. 2; 9.
23;
10. 400;
11. 7;
12. 28;
13. 20;
14. 13; 15.
12; 16. 30; 17. 5; 18.
18; 19. 8; 20. 2; 21.
8; 22. 2; 23. 8;
24. 4; 25. 64; 26. 1; 27.
36; 28. 125; 29. 625; 30.
27; 31. 20; 32. 52;
33. 15;
34. 20;
35. 54;
36. 30;
37. 20;
38. 21; 39.
38; 40. 5; 41. 127; 42.
54; 43. 64; 44. 125; 45.
2; 46. 1; 47. 5; 48.
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