3.1.2. Zero and Negative Exponents
In Lesson 3.1.1, you used the meaning of an exponent to rewrite expressions such as y4 · y 2 and (x2y)3. Today
you will use the patterns you discovered to learn how to interpret expressions with exponents that are
negative or zero.
3-12. Review what you learned about exponents in Lesson 3.1.1 to rewrite each expression below as simply as
possible. If you see a pattern or know of a shortcut, be sure to share it with your teammates.
1.
x7 · x 4
2.
(x3)3
4.
(x2y 2)4
5.
3.
6.
3-13. With your study team, summarize the patterns you found in problem 3-12. For each one, simplify the
given expression and write an expression that represents its generalization. Then, in your own words, explain
why the pattern works.
a.
Expression
Generalization
x25 · x40 = ?
xm · x n = ?
(x5)12 = ?
(xm)n = ?
b.
c.
Why is this true?
3-14. Describe everything you know about
by answering the following:
a. What is its value?
b. How can you rewrite it using a single exponent?
c. What new conclusions can you draw? Be prepared to explain your findings to the class.
3-15. Problem 3-14 helped you recognize that x0 = 1. Now you will similarly use division to explore the
meaning of x−1, x−2, etc. Simplify each of the expressions below twice: 1. Once by expanding the terms and
simplifying. 2. Again by using your new pattern for division with exponents. Be ready to discuss the meaning of
negative exponents with the class.
a.
1 expanding the terms and simplifying.
2. Again by using your new pattern for division with exponents.
b.
1 expanding the terms and simplifying.
2. Again by using your new pattern for division with exponents.
c.
1 expanding the terms and simplifying.
2. Again by using your new pattern for division with exponents.
Laws of Exponents
In the expression x3, x is the base and 3 is the exponent.
x3 = x · x · x
The patterns that you have been using during this section of the book are called the laws of exponents. Here
are the basic rules with examples
3-16. Use your exponent patterns to rewrite each of the expressions below. For example, if the original
expression has a negative exponent, then rewrite the expression so that it has no negative exponents, and vice
versa. Also, if the expression contains multiplication or division, then use your exponent rules to simplify the
expression.
a. k−5
e. y-2/y-3
b. m0
f.
c. x−2 · x5
g. (a2b)−1
d.
h.
(x−2)3
3-17. EXPONENT CONCENTRATION
Split your team into two pairs and decide which is Team A and which is Team B. Your teacher will
distribute a set of cards for a game described below.
o
o
o
o
o
Arrange the cards face down in a rectangular grid.
Team A selects and turns over two cards.
If Team A thinks the values on the cards are equivalent, they must justify this claim to Team
B. If everyone in Team B agrees, Team A takes the pair. If the values are not equivalent, Team
A returns both cards to their original position (face down). This is the end of the turn for Team
A.
Team B repeats the process.
Teams alternate until no cards remain face down. The team with the most matches wins.
3-18. Below, describe the meaning of zero and negative exponents. That is, explain how to
interpret x0 and x−1.
“Zero and Negative Exponents”
HW 18 3-19 – 3-24