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1
1.1.2: Rational and Irrational Numbers and Their Properties
Assignment #: pg. 15
1. Answer for #1
2. Answer for #4
3. Answer for #5
4. Answer for #6
Test corrections due:
Oct. 8
Three parts needed for EVERY Problem:
1. State the problem you missed
2. REDO the problem (must show your work)
3. In 1-2 sentences, tell me WHY you missed
that problem.
3
Pg. 19 in FOLDER
Read the problem.
What do we know?
What are we looking for?
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1.1.2: Rational and Irrational Numbers and Their Properties
Key Concepts, continued
Properties of Exponents
Words
Symbols
Zero Exponent
Property
A base raised to the
power of 0 is equal to 1.
a0 = 1
Negative Exponent
Property
A negative exponent of a
number is equal to the
reciprocal of the positive
exponent of the number.
a
-
m
n
=
Numbers
120 = 1
1
,
m
a
a ¹ 0, n ¹ 0
n
64
1.1.1: Defining, Rewriting, and Evaluating Rational Exponents
-
2
3
=
1
2
64
3
=
1
16
5
Key Concepts, continued
Words
Product of Powers
Property
To multiply powers
with the same base,
add the exponents.
Quotient of
Powers Property
To divide powers
with the same base,
subtract the
exponents.
Symbols
Numbers
1
a ·a = a
m
n
a
m+n
a
1
34 · 34 = 34
+
7
4
=
32 = 9
4
m
n
7
=a
m-n
89
1
4
= 89
-
1
1
9
= 83 = 2
89
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1.1.1: Defining, Rewriting, and Evaluating Rational Exponents
Key Concepts, continued
Words
Power of a Power
Property
To raise one power
to another power,
multiply the
exponents.
Power of a
Product Property
To find the power of
a product, distribute
the exponent.
Symbols
Numbers
(a )
m
n
= am · n
3
2
æ ö
·3
3
3
=
ç5 ÷ = 5
è ø
2
52 = 25
( ab)
m
= amb m
( 25 · 36)
1
1
2
=
1
25 2 · 36 2 =
5 · 6 = 30
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1.1.1: Defining, Rewriting, and Evaluating Rational Exponents
Key Concepts, continued
Words
Power of a
Quotient Property
To find the power of
a quotient, distribute
the exponent.
Symbols
m
æ aö
a
ç b ÷ = bm
è ø
m
Numbers
1
1
æ 25 ö 2 25 2 5
ç 49 ÷ = 1 = 7
è ø
49 2
• Either the power or root can be determined first when
evaluating an exponential expression with a rational
exponent.
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1.1.1: Defining, Rewriting, and Evaluating Rational Exponents
Key Concepts, continued
• The sum of two rational numbers is a rational number.
• The product of two rational numbers is a rational
number.
• The sum of a rational number and an irrational
number is an irrational number.
• The product of a rational number and an irrational
number is an irrational number.
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1.1.2: Rational and Irrational Numbers and Their Properties
Guided Practice
Example 1
6
3
Simplify the expressiona 5 · a 2 .
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1.1.2: Rational and Irrational Numbers and Their Properties
Guided Practice: Example 1, continued
1. Identify which property can be used to
simplify the expression.
This is the product of two exponential expressions
with the same base. Use the Product of Powers
Property to simplify.
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1.1.2: Rational and Irrational Numbers and Their Properties
Guided Practice: Example 1, continued
2. Apply the property to simplify the
expression.
The Product of Powers Property states that if the
bases are the same, the expression can be written as
the single base raised to the sum of the powers.
6
3
6
a5 · a2 = a5
+
3
27
2
= a10
✔
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1.1.2: Rational and Irrational Numbers and Their Properties
Guided Practice
Example 2
7
Simplify the expression
b9
8
.
b3
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1.1.2: Rational and Irrational Numbers and Their Properties
Guided Practice: Example 2, continued
1. Identify which property can be used to
simplify the expression.
This is the quotient of two exponential expressions
with the same base. Use the Quotient of Powers
Property to simplify.
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1.1.2: Rational and Irrational Numbers and Their Properties
Guided Practice: Example 2, continued
2. Apply the property to simplify the
expression.
7
b9
8
b3
7
=b
9
-
8
3
=b
-
17
9
or
1
17
b9
✔
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1.1.2: Rational and Irrational Numbers and Their Properties