Lesson 8.2.3
HW: Day 1: problems 8-88 to 8-93
Day 2: problems 8-94 to 8-99
Learning Target: Scholars will continue to develop methods for simplifying expressions with positive
exponents, and will learn what negative and zero exponents represent.
Earlier in this chapter you learned how to write large numbers in scientific notation. Astronomers use
those large numbers to measure great distances in space. Not all scientists work with such a large scale,
however. Some scientists use very small numbers to describe what they measure under a microscope. In
this lesson, you will continue your work with exponents, and then you will turn your attention to using
scientific notation to represent small numbers.
8-72. Two of the problems below are correct, and four contain errors. Expand each original expression to
verify that it is correct. If it is not, identify the mistake and simplify to find the correct answer.
1.
2.
3.
4.
5.
6.
8-73. Rewrite each expression in a simpler form using the patterns you have found for rewriting
expressions with exponents. If it is reasonable, write out the factored form to help you.
1. 23 · 24
2.
3. (5x2)3
4. (4x)2(5x2)
8-75. Salvador was studying microscopic pond animals in science class. He read that amoebas were 0.3
millimeters to 0.6 millimeters in length. He saw that euglenas are as small as 8.0 · 10−2 millimeters, but
he did not know how big or small a measurement that was. He decided to try to figure out what a
negative exponent could mean.
Copy and complete Salvador’s calculations at right. Use the pattern of
dividing by 10 to fill in the missing values.
2. How does 102 related to 10–2?
3. What type of numbers did the negative exponents create? Did negative exponents create negative
numbers?
1.
8-76. Ngoc was curious about what Salvador was doing and began exploring patterns, too. He completed
the calculations at right.
1. Copy and complete his calculations. Be sure to include all of the integer exponents from 5 to –3.
2. Look for patterns in his list. How are the values to the right of the equal sign changing? Is there
a constant multiplier between each value?
8-77. In problems 8-75 and 8-76, you saw that 100 and 20 both simplify to the same value. What is it? Do
you think that any number to the zero power would have the same answer? Explain.
8-78. Both Ngoc and Salvador are looking for ways to calculate values with negative exponents without
extending a pattern. Looking at the expression 5−2, they each started to simplify differently.
Salvador thinks that 52 is 25, so 5−2 must be
. Ngoc thinks 5−2 is
. Which student is correct?
8-79. Salvador’s first questions about negative exponents came from science class, where he had learned
that euglenas measured 8.0 × 10−2 millimeters. Use your understanding of negative exponents to
rewrite 8.0 × 10−2 in standard form.
8-80. The probability of being struck by lightening in the United States is 0.000032%, while the
probability of winning the grand prize in a certain lottery is 6.278 × 10−7 percent. Which event is more
likely to happen? Explain your reasoning, including how you rewrote the numbers to compare them.
8-81. The list below contains a star, a river, and a type of bacteria. Which one is which? Use your
understanding of scientific notation to put the three items listed below in order from largest to smallest
and identify which is which.
1. Yangtze measures 6.3 × 105 meters.
2. Staphylococcus measures 6 × 10−7 meters.
3. Eta Carinae measures 7.1 × 1017meters.
8-82. Which of the numbers below are correctly written in scientific notation? For each that is not,
rewrite it correctly.
1. 4.51 × 10−2
2. 0.789 × 105
3. 31.5 × 102
4. 3.008 × 10−8
8-83. Create a fraction from these expressions, and then show how you use a Giant One to simplify.
1. 65 · 6−3
2. w5w−2
3. 10−4 · 105
8-84. Now it is time to reverse your thinking. If negative exponents can create fractions, then can
fractions be written as expressions with negative exponents? Simplify the expressions below. Write your
answer in two different forms: as a fraction and as an expression with a negative exponent.
1.
2.
3.
8-88. Decide which numbers below are correctly written in scientific notation. If they are not, rewrite
them.
1.
2.
3.
4.
92.5 × 10−2
6.875 × 102
2.8 × 10
0.83 × 1002
8-89. In the table below, write each power of 10 as a decimal and as a fraction.
5. Describe how the decimals and fractions change as you progress down the table.
6. How would you tell someone how to write 10−12 as a fraction? You do not have to write
the actual fraction.
Power of 10 Decimal Form Fraction Form
100
10–1
0.1
10–2
10–3
10–4
10–5
8-90. Mary wants to have $8500 to travel to South America when she is 21. She currently has $6439 in a
savings account earning 4% annual compound interest. Mary is 13 now.
7. If Mary does not take out or deposit any money, how much money will Mary have when
she is 15?
8. Will Mary have enough money for her trip when she is 21?
9. If Mary were to graph this situation, describe what the graph would look like.
8-91. Recall that vertical lines around a number are the symbol for the absolute value of a
number. Simplify each expression.
10.
11.
12.
13.
14.
15.
8-92. For each equation below, solve for x. Sometimes the easiest strategy is to use mental math.
16. x − = 1
17. 5.2 + x = 10.95
18. 2x − 3.25 = 7.15
19.
8-93. Determine the coordinates of each point of intersection without graphing.
20. y = 2x − 3
y = 4x + 1
21. y = 2x − 5
y = −4x −2
8-94. Write each number in scientific notation.
22. 5467.8
23. 0.0032
24. 8,007,020
8-95. Simplify each expression using the rules for exponents.
25.
26. 10x4(10x)−2
27. (
)3 · (4)2
28.
8-96. Simplify each expression.
29.
30.
31.
32.
33.
34.
8-97. Athletes in the Middle Plains School District regularly receive personal advising on their
nutrition. Coaches wondered if the nutritional advising was having an impact, so they divided athletes
into two groups. One group received advice and one group did not. After six months, they collected the
following data:
35. Which is the independent variable?
36. Make a relative frequency table.
37. Is there an association between receiving the nutritional advice and regularly eating a
balanced breakfast?
8-98. Graph the points X(−3, 5), Y(−2, 3), and Z(−1,4). Connect them to make a triangle. 8-98 HW eTool
(Desmos).
38. Reflect the triangle across the y-axis. What are the new coordinates of point Z?
39. Translate the original triangle down 6 units and right 3 units. What are the new
coordinates of point Y?
40. Dilate the original triangle by multiplying each coordinate by −1. Describe the new
shape you create.
8-99. The table below shows the amount of money Francis had in his bank account each day since he
started his new job. 8-99 HW eTool (Desmos).
41. Write a rule for the amount of money in Francis’s account. Let x represent the number
of days and y represent the number of dollars in the account.
42. When will Francis have more than $1000 in his account?
Lesson 8.2.3


8-72.
1.
2.
3.
4.
See below:
Correct
Multiplied exponents instead of adding, 27
Correct
Multiplied bases, 37
5.
6.
8-73.
1.
2.
3.
4.
Eliminated numerator,
The difference of the exponents is zero, but the quantities divide to yield 1.
See below:
27
23
(5x2)(5x2)(5x2) = 53x6
(4x)(4x)(5x · x) = 425x4 = 80x4
8-75. See below:
1. 100 = 1, 10−1 =
, 10−2 =
, 10−3=
2. 102 is 100 and 10−2 is
, so they are inverses of each other.
3. Negative exponents create decimals or fractions. No.
 8-76. See below:
1. 22 = 4, 21 = 2, 20 = 1, 2−2 = , 2−3 =
2. The values are being divided by two as you work down the list. The multiplier from one
value to the next value in the list would be
.
 8-77. 1. All numbers except 0 have a value of 1 when raised to the zero power.
 8-78. Both boys are correct and get the same answer of
.
 8-79. 0.08 millimeters
 8-80. The probability of being struck by lightening is 3.2 · 10-5, which is greater than that of
winning the lottery.
 8-81. Eta Carinae (a star) is largest, Yangtze (river) is middle, staphylococcus (bacteria) is
smallest.
 8-82. See below:
1.
2.
3.
4.
correct
7.89 × 104
3.15 × 103
correct
 8-83. See below:
1.
2.
3.
 8-84. See below:
1.
, 6−2
2.
, m−1
3.

, 10−3
8-88.
1.
2.
3.
4.
See below:
9.25 × 10−1
correct
correct
8.3 × 103

8-89. See answers in table.
1. The 1 moves one decimal place to the right, and the denominator of the fraction is
multiplied by 10.
2. Write 1 in the numerator and 1 followed by 11 zeros in the denominator.

8-90.
1.
2.
3.
See below:
$6964.42
yes; $8812.22
The graph would be a curve that begins at $6439 and curves upwards.

8-91.
1.
2.
3.
4.
5.
6.
See below:
6
17
–4.5
3
13
–4

8-92.
1.
2.
3.
4.
8-93.
1.
See below:
x=2
x = 5.75
x = 5.2
x=6
See below:
(−2, −7)


2. ( , −4)
8-94. See below:
1. 5.4678 × 103


2. 3.2 × 10−3
3. 8.00702 ×1 06
8-95. See below:
1.
= 3−5
2.
= 10−1x2
3.
4. x2
8-96. See below:
1.
or
2.
3.
or
4.
5. –4


6.
or 10
8-97. See below:
1. Receiving nutrition advice.
2. See table below
3. There does not appear to be an association. Only about 35% of athletes eat a
balanced breakfast, regardless of whether they received the nutritional advice or
not.
8-98. See below:

1. (1, 4)
2. (1, −3)
3. The new shape is the same size as the original. It is rotated 180° or reflected over
the x-axis and the y-axis.
8-99. See below:
1. y = 27 + 43x
2. After the 23rd day.