Topic 9: Negative Exponents
for use after
Accentuate the Negative
Investigation 4
You have used positive exponents to represent repeated multiplication.
Negative exponents are used to represent repeated division.
A. You can use patterns to find the value of negative exponents.
1. Copy and complete the table below.
2. What pattern do you see as you go from 2
4
to 2
3
and from
2
3
to 2
2
?
3. The value of 2
8
is 256. Explain how to find the value of 2
7
using
division.
B. 1. Extend the table in part (A). Find 2
1
,2
0
,2
-1
,2
-2
,2
-3
, and 2
-4
.
Express values less than 1 as fractions.
2. You can use multiplicative inverses to find the value of an
expression with a negative exponent. Use your answers in
Part A (4) and Part B (1) to complete each statement.
2–
2
=
■
2–
3
=
■
2–
4
=
■
3. Rewrite each expression using its reciprocal and a positive
exponent. Then find the value of the expression.
a. 4
-2
b.
Q
2
-3
5
R
Q
1
2
R
Q
1
2
R
Q
1
2
R
Expression
Value
2
4
2
3
2
2
Topic 9: Negative Exponents
Teaching Guide
Mathematical Goals
•
Simplify expressions containing negative exponents
At a Glance
Negative exponents are difficult for many students to understand. Some
students may simply write a negative sign in front of their solution. It is
necessary to explain to the students why negative exponents represent
repeated division.
Positive exponents have already been covered in this course. A class
discussion to review this topic would be beneficial at this time. By reviewing
the fact that positive exponents indicate repeated multiplication, it will
become apparent to students why negative exponents indicate repeated
division. For example, since 2
3
= 2 3 2 3 2 and 2
2
= 2 3 2, decreasing the
exponent by 1 is equivalent to dividing by the base, 2. At this point, it may
also be helpful to review the fact that any base raised to the zero power is
equal to 1. Therefore, negative exponents involve repeated division.
After Problem 9.1A, have students choose another number and make a
table of the values of that number raised to powers from 3 to 23.
Summarize Problem 9.1B by asking:
•
In what other mathematical situations do we use reciprocals?
•
Why does it make sense to use reciprocals when finding the value of an
expression with a negative exponent?
Homework Check
When reviewing Exercise 7, ask:
•
Does each method work for solving the problem? What are some
advantages and disadvantages of each method?
Summarize Exercises 14–19 by asking:
•
How is knowing powers of 10 useful in measurement?
•
How could some of these exercises be used to describe metric units?
PACING
1 day
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Assignment Guide for Topic 9
Core
1–19
Advanced
20
Answers to Topic 9
Problem 9.1
A. 1.
16;8;4
2.
Answers may vary. Sample: 2
3
is half of 2
4
,
and 2
2
is half of 2
3
. Each term is one half of
the previous term.
3.
2
8
5 2
7
3 2,so 2
7
5 2
8
4 2;divide 256
by 2.
B. 1.
2, 1, , , ,
2.
2; 3; 4
3. a.
=
b.
=
Exercises
1.
2.
3.
4.
5.
6.
7.
Yes; dividing by 3 twice is the same
as multiplying by .
8.
9.
100
10.
196
11.
4
12.
13.
96
14.
200
15.
0.02
16.
0.007
17.
7,000
18.
250,000
19.
0.000025
20. a.
625; 125; 25; 5; 1
b.
10,000; 1,000; 100; 10; 1
c.
The value of any positive number with
exponent zero is equal to 1.
5
81
1
8
1
9
2
2
49
27
512
5
2
1
125
1
243
1
2
125
Q
5
8
2
R
3
1
Q
1
16
4
R
2
1
16
1
8
1
4
1
2
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