Download Key Concepts. Rational Exponents

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Lesson 7-4 (pp. 379–384)
Lesson Objective
1 Simplifying expressions with rational
exponents
Rational Exponents
NAEP 2005 Strand: Algebra
Topic: Variables, Expressions, and Operations
Local Standards: ____________________________________
Key Concepts.
Rational Exponents
If the nth root of a is a real number and m is an integer, then
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All rights reserved.
1
an n
m
n
a and a n
m
m
n
a
(a )
If m is negative, a 0.
Properties of Rational Exponents
Let m and n represent rational numbers. Assume that no denominator equals 0.
Property
Example
1
2
1 12
3
am ? an am n
83 ? 83 5 83
(am)n a mn
(512)4 512 ? 4 52 25
(ab)m a m b m
(4 ? 5)2 42 ? 52 2 ? 52
(am) a
1
m
am am n
an
m
a m a
b
m
b
()
1
1
5 81 5 8
1
1
1
1
922 912 13
3
p2 32 2 12 π 1 π
p
1
p2
( )
5
27
1
3
1
1
3
3
5 1 53
273
Another way to write a radical expression is to use a rational exponent.
Like the radical form, the exponent form always indicates the principal root.
A rational exponent may have a numerator other than 1.
All of the properties of integer exponents also apply to rational exponents.
You can simplify a number with a rational exponent by using the properties of exponents or
by converting the expression to a radical expression.
To write an expression with rational exponents in simplest form, write every exponent as a
positive number.
Algebra 2 Daily Notetaking Guide
Lesson 7-4
141
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Examples.
1 Simplifying Expressions With Rational Exponents Simplify each expression.
1
a. 643
1
64 3 3
64
4
3
Rewrite as a radical.
3
Rewrite 64 as a cube.
3
1
cube
Definition of
root.
1
b. 7 2 ? 7 2
1
7
?
7
Rewrite as radicals.
7
c. 5
5
1
3
1
3
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1
72 ? 72 By definition,7 is the number whose square is 7 .
1
25 3
?
1
3
5 ? 3
5 ?
? 25 3 3
Rewrite as radicals.
25
25
Property for multiplying radical expressions.
3
5
By definition, 5 is the number whose cube is
5 .
2 Converting To and From Radical Form
x
2
7
7
x
2
0.4
in radical form.
7
or (x)
2
y
0.4
y
2
5
1
1
or
5
y
b. Write the radical expressions
c
4
3
3
c
4
3
3
(5 y)
2
2
5
and (b) in exponential form.
5
c 4
(3 b)5 b
3
3 Simplifying Numbers With Rational Exponents Simplify 252.5.
Method 1
25
2.5
5
(
25 2 5 2
5
142
Method 2
Lesson 7-4
5 1
5 5
(
( )
5
2
5
2
5
2
25
2.5
5
25 2 1
3125
1
5 5
1
5
25 2
1
(25 )
5
1
3125
Algebra 2 Daily Notetaking Guide
© Pearson Education, Inc., publishing as Pearson Prentice Hall.
2
a. Write x 7 and y
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2
4 Writing Expressions in Simplest Form Write (243a
( )
( )
2
2
10
(243a ) 5
(3
5
2
10
a )5
3
5
5
10 5
)
in simplest form.
2
? a
(10)
5
3
2
4
a
3
a
2
9
4
a
4
Check Understanding.
1. Simplify each expression.
1
1
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a. 164
1
3
2. a. Write the expressions y28 and z0.4 in
radical form.
1
c. 22 ? 82
2
2
1
5 2
8 3 ; "z
"
y
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1
b. 22 ? 22
4
3
b. Write the expressions "x2 and (!y) 3 in
exponential form.
2
3
x3 ; y2
c. Critical Thinking Refer to the definition of rational exponents. Explain the
need for the following restriction: If m is negative, a 0.
If m is negative, then a is in the radicand in the denominator of a
fraction. If a 0, then the denominator would be zero. Since this
cannot happen, a 0.
3. Simplify each number.
232
3
b. 325
a. 25
1
125
4. Write (8x 15)
8
213
4
c. (232) 5
16
in simplest form.
1
2x5
Algebra 2 Daily Notetaking Guide
Lesson 7-4
143