section concepts 2.4 definitions of b 0 and b –n slide 1 copyright (c) the mcgraw-hill companies,...
TRANSCRIPT
Section
Concepts
2.4 Definitions of b0 and b–n
Slide 1Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
1. Definition of b0
2. Definition of b–n 3. Properties of Integer Exponents: A Summary
Section
Concepts
2.4 Definitions of b0 and b–n
Slide 2Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Any Homework Questions?
Section 2.4 Definitions of b0 and b–n
1. Definition of
Slide 3Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Consider the following pattern.
As the exponents decrease by1, the resulting expressions are divided by 3.
For the pattern to continue, we define
DEFINITION
Let b be a nonzero real number. Then, b0 = 1.
Definition of b0
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Example 1 Simplifying Expressions with a Zero Exponent
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Simplify (assume
a.
d.
b.
e.
c.
f.
Section 2.4 Definitions of b0 and b–n
1. Definition of b0
Slide 6Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Therefore, must be defined as 1.
Section 2.4 Definitions of b0 and b–n
2. Definition of b–n
Slide 7Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Consider the following pattern.As the exponents decrease by1, the resulting expressions aredivided by 3.
For the pattern to continue, we
define
Section 2.4 Definitions of b0 and b–n
2. Definition of b–n
Slide 8Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
For the pattern to continue, we define
For the pattern to continue, we define
DEFINITION
Let n be an integer and b be a nonzero real number. Then,
Definition of b–n
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Section
The definition of implies that to evaluatetake the reciprocal of the base and change the sign of the exponent.
2.4 Definitions of b0 and b–n
2. Definition of b–n
Slide 10Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Example 2 Simplifying Expressions with Negative Exponents
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Simplify. Assume that
a. b. c.
Example 3 Simplifying Expressions with Negative Exponents
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Simplify. Assume that
a. b. c.
Example 4 Simplifying Expressions with Negative Exponents
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Simplify. Assume that
a. b. c.
Section 2.4 Definitions of b0 and b–n
3. Properties of Integer Exponents: A Summary
Slide 14Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Properties of Integer ExponentsAssume that a and b are real numbers and that m and n represent integers.
Section 2.4 Definitions of b0 and b–n
3. Properties of Integer Exponents: A Summary
(continued)
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Section 2.4 Definitions of b0 and b–n
3. Properties of Integer Exponents: A Summary
(continued)
Slide 16Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Section 2.4 Definitions of b0 and b–n
3. Properties of Integer Exponents: A Summary
Slide 17Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Example 5 Simplifying Expressions with Exponents
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Simplify the expressions. Write the answers with positive exponents only. Assume all variables are nonzero.
a. b. c.
Example 6 Simplifying Expressions with Exponents
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Simplify the expressions. Write the answers with positive exponents only. Assume that all variables are nonzero.
a. b. 4 2 2 12 5p q p q 22ss
Example 7 Simplifying Expressions with Exponents
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Simplify the expressions. Write the answers with positive exponents only. Assume that all variables are nonzero.
a. b.
Example 8 Simplifying Expressions with Exponents
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Simplify the expressions. Write the answers with positive exponents only. Assume that all variables are nonzero.
a. b.
43 2
24 63
s t
s t
3 3 2
92
4
xyxy
x y
Example 9 Simplifying an Expression with Exponents
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Simplify the expressionWrite the answer with positive exponents only.
Example2.3
Definitions of b0 and b–n
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1. Definition of b0
2. Definition of b–n 3. Properties of Integer Exponents: A Summary
Example
You Try:
1. Simplify each expression. Write the answers with positive exponents only. Assume all variables are nonzero.
A. B.2
3 4
x
y z
4 2
2 2
35
21
x y
x y
Example
You Try:
1. Simplify each expression. Write the answers with positive exponents only. Assume all variables are nonzero.
A. B. 33 02a b
12 16 28 2y y z