0.3
Exponents and the Order
of Operations
0.3
OBJECTIVES
1. Write a product of factors in exponential form
2. Evaluate an expression involving several operations
Often in mathematics we define symbols that allow us to write a mathematical statement in
a more compact or “shorthand” form. This is an idea that you have encountered before. For
example, the repeated addition:
555
NOTE
5 5 5 15
and
3 5 15
can be rewritten as
35
Thus multiplication is shorthand for repeated addition.
In algebra, we frequently have a number or variable that is repeated as a factor in an
expression several times. For instance, we might have
555
NOTE A factor is a number or a
To abbreviate this product, we write
variable that is being multiplied
by another number or variable.
5 5 5 53
This is called exponential notation or exponential form. The exponent or power, here 3,
indicates the number of times that the factor or base, here 5, appears in a product.
Exponent or power
C A U TI O N
Be careful: 53 is not the same as
5 3. Notice that
53 5 5 5 125 and
5 3 15.
5 5 5 53
Factor or base
This is read “5 to the third power” or “5 cubed.”
Example 1
Writing in Exponential Form
Write 3 3 3 3, using exponential form. The number 3 appears 4 times in the product, so
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Four factors
of 3
3 3 3 3 34
This is read “3 to the fourth power.”
CHECK YOURSELF 1
Rewrite each using exponential form.
(a) 4 4 4 4 4 4
(b) 7 7 7 7
29
AN ARITHMETIC REVIEW
C AU TI ON
Only one of these results can be
correct.
To evaluate an arithmetic expression, you need to know the order in which the operations
are done. To see why, simplify the expression 5 2 3.
Method 1
or
Method 2
523
523
Add first.
Multiply first.
CHAPTER 0
30
73
21
56
11
Because we get different answers depending on how we do the problem, the language of
mathematics would not be clear if there were no agreement on which method is correct.
The following rules tell us the order in which operations should be done.
Step by Step:
NOTE Parentheses and
brackets are both grouping
symbols. Later we will see that
fraction bars and radicals are
also grouping symbols.
Step 1
Step 2
Step 3
Step 4
The Order of Operations
Evaluate all expressions inside grouping symbols first.
Evaluate all expressions involving exponents.
Do any multiplication or division in order, working from left to right.
Do any addition or subtraction in order, working from left to right.
Example 2
Evaluating Expressions
Evaluate 5 2 3.
There are no parentheses or exponents, so start with step 3: First multiply and then add.
523
Multiply first.
56
NOTE Method 2 shown above
is the correct one.
Then add.
11
CHECK YOURSELF 2
Evaluate the following expressions.
(a) 20 3 4
(b) 9 6 3
When there are no parentheses, evaluate the exponents first.
Example 3
Evaluate 5 32.
5 32 5 9
Evaluate the power first.
45
CHECK YOURSELF 3
Evaluate 4 24.
© 2001 McGraw-Hill Companies
© 2001 McGraw-Hill Companies
Evaluating Expressions
EXPONENTS AND THE ORDER OF OPERATIONS
SECTION 0.3
31
Both scientific and graphing calculators correctly interpret the order of operations. This
is demonstrated in Example 4.
Example 4
Using a Calculator to Evaluate Expressions
Use your scientific or graphing calculator to evaluate each expression. Round the answer to
the nearest tenth.
(a) 24.3 6.2 3.53
When evaluating expressions by hand, you must consider the order of operations. In
this case, the multiplication must be done first, then the addition. With a calculator, you
need only enter the expression correctly. The calculator is programmed to follow the
order of operations.
Entering 24.3
NOTE With most graphing
calculators, the final command
is ENTER . With most other
scientific calculators; the key is
marked .
6.2
3.53
ENTER
yields the evaluation 46.186. Rounding to the nearest tenth, we have 46.2.
(b) 2.453 49 8000 12.2 1.3
Some calculators use the carat (^) to designate powers. Others use the symbol xy
(or y x ).
3 49 8000 12.2 1.3
Entering 2.45 ^
or
2.45 y
x
ENTER
3 49 8000 12.2 1.3 yields the evaluation 30.56. Rounding to the nearest tenth, we have 30.6.
CHECK YOURSELF 4
Use your scientific or graphing calculator to evaluate each expression.
(a) 67.89 4.7 12.7
(b) 4.3 55.5 3.753 8007 1600
Operations inside grouping symbols are done first.
Example 5
Evaluating Expressions
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Evaluate (5 2) 3.
Do the operation inside the parentheses as the first step.
(5 2) 3 7 3 21
Add.
CHECK YOURSELF 5
Evaluate 4 (9 3).
The principle is the same when more than two “levels” of operations are involved.
AN ARITHMETIC REVIEW
Example 6
(a) Evaluate 4 (2 3)3.
Add inside the parentheses first.
4 (2 3)3 4 (5)3
Evaluate the power.
4 125
Multiply.
500
(b) Evaluate 5 (7 3)2 10.
Evaluate the expression inside the parentheses.
5 (7 3) 10 5(4)2 10
2
Evaluate the power.
5 16 10
Multiply.
80 10 70
Subtract.
CHECK YOURSELF 6
Evaluate.
(a) 4 33 8 11
(b) 12 4 (2 3)2
CHECK YOURSELF ANSWERS
1. (a) 46; (b) 74
6. (a) 20; (b) 112
2. (a) 8; (b) 11
3. 64
4. (a) 8.2; (b) 190.92
5. 24
© 2001 McGraw-Hill Companies
CHAPTER 0
32
Name
Exercises
0.3
Section
Date
Write each expression in exponential form.
ANSWERS
1. 7 7 7 7
1.
2. 2 2 2 2 2 2
2.
3.
3. 6 6 6 6 6
4.
4. 4 4 4 4 4 4 4
5.
6.
5. 8 8 8 8 8 8 8 8 8 8
7.
6. 10 10 10
8.
9.
7. 15 15 15 15 15 15
10.
8. 31 31 31 31 31 31 31 31 31 31
11.
12.
Evaluate each of the following expressions.
13.
9. 7 2 6
10. 10 4 2
11. (7 2) 6
12. (10 4) 2
13. 12 8 4
14. 10 20 5
15. (12 8) 4
16. (10 20) 5
14.
15.
16.
17.
18.
© 2001 McGraw-Hill Companies
19.
17. 8 7 2 2
18. 48 8 4 2
19. 8 (7 2) 2
20. 48 (8 4) 2
21. 3 5
22. 5 2
23. (3 5)2
24. (5 2)3
20.
21.
22.
2
3
23.
24.
25.
25. 4 32 2
26. 3 24 8
26.
33
ANSWERS
27.
28.
29.
27. 7 (23 5)
28. 4 (32 7)
29. 3 24 6 2
30. 4 23 5 6
31. (2 4)2 8 3
32. (3 2)3 7 3
33. 4 (2 6)2
34. 3 (8 4)2
35. (4 2 6)2
36. (3 8 4)2
37. 3 (4 3)2
38. 5 (4 2)3
39. 3 4 32
40. 5 4 23
41. 4 (2 3)2 25
42. 8 + 2 (3 + 3)2
43. (4 2 3)2 25
44. 8 (2 3 3)2
30.
31.
32.
33.
34.
35.
36.
37.
Evaluate using your calculator. Round your answer to the nearest tenth.
38.
45. (1.2)3 2.0736 2.4 1.6935 2.4896
39.
46. (5.21 3.14 6.2154) 5.12 .45625
40.
47. 1.23 3.169 2.05194 (5.128 3.15 10.1742)
41.
48. 4.56 (2.34)4 4.7896 6.93 27.5625 3.1269 (1.56)2
42.
49. Population doubling. Over the last 2000 years, the Earth’s population has doubled
approximately 5 times. Write this last factor in exponential form.
43.
50. Volume of a cube. The volume of a cube with each edge of length 9 in. is given by
44.
9 9 9. Write the volume using exponential notation.
Insert grouping symbols in the proper place so that the given value of the expression is
obtained.
45.
46.
51. 36 4 2 4; 2
52. 48 3 2 2 3; 2
47.
53. 6 9 3 16 4 2; 29
48.
50.
Answers
51.
1. 74
3. 65
5. 810
7. 156
9. 19
11. 54
13. 10
15. 1
17. 60
19. 144
21. 75
23. 225
25. 34
27. 21
29. 36
31. 40
33. 256
35. 196
37. 147
39. 21
41. 75
43. 96
45. 1.2
47. 7.8
49. 25
51. 36 (4 2) 4
53. (6 9) 3 (16 4) 2
52.
53.
54.
34
© 2001 McGraw-Hill Companies
54. 5 3 2 8 5 2; 28
49.