Download Properties of Rational Exponents

7.2
Properties of Rational Exponents
Goals p Use properties of rational exponents.
p Use properties of rational exponents in real life.
Your Notes
VOCABULARY
Simplest form of a radical A radical expression after you
apply the properties of radicals, remove any perfect
nth powers, and rationalize any denominators
Like radicals Two radical expressions that have the
same index and the same radicand
PROPERTIES OF RATIONAL EXPONENTS
Let a and b be real numbers and let m and n be rational
numbers. The following properties have the same names
as those listed in Lesson 6.1, but now apply to rational
exponents as illustrated.
Property
Example
1. am p an am n
31/2 p 33/2
3(1/2 3/2) 32 9
2. (am)n amn
(43/2)2 4(3/2 p 2) 43 64
3. (ab)m ambm
(9 p 4)1/2
91/2 p 41/2 3 p 2 6
1
4. am m
, a 0
a
1
1
251/2 1/2
25
5
am
5. am n, a 0
an
65/2
1
6(5/2 1/2) 62 36
6 /2
a m
am
6. 冢冣 m
, b 0
b
b
8 1/3
81/3
2
冢 27 冣
1
/
3
27
3
Lesson 7.2 • Algebra 2 Notetaking Guide
155
Your Notes
Using Properties of Rational Exponents
Example 1
Use properties of rational exponents to simplify the
expression.
a. 72/3 p 71/2 7(2/3 1/2) 77/6
b. (31/4 p 51/3)3 (31/4)3 p (51/3)3 (31/4 p 3) p (51/3 p 3)
33/4 p 5
101/2
10(1/2 1/4) 101/4
c. 1
/
4
10
121/6
d. 1
3 /6
冢
4
冣
冤冢132 冣 冥
1/6 4
(41/6)4 4(1/6 p 4) 42/3
Using Properties of Radicals
Example 2
Use the properties of radicals to simplify the expression.
5
5
a. 兹苶
16 p 兹2
苶
3
08
兹1苶苶
b. 3
兹4苶
5
5
兹苶
16 p 2 兹32
苶 2
冪莦
3
108
4
3
兹27
苶 3
Use the product
property.
Use the quotient
property.
Checkpoint Simplify the expression.
4
1.
(65
p
8
兹4苶
2. 4
兹3苶
25)1/5
1
12
2
Writing Radicals in Simplest Form
Example 3
Write the expression in simplest form.
4
4
兹苶
48 兹苶
16 p 3
4
4
兹苶
16 p 兹3
苶
4
2兹3
苶
156
Algebra 2 Notetaking Guide • Chapter 7
Factor out perfect fourth power.
Product property
Simplify.
Your Notes
Adding and Subtracting Roots and Radicals
Example 4
Perform the indicated operation.
a. 3(53/4) 4(53/4) (3 4)(53/4) 7(53/4)
4
4
4
4
b. 兹64
苶 兹4
苶 兹苶
16 p 4 兹4
苶
4
4
4
兹苶
16 p 兹4
苶 兹4
苶
4
4
2兹4
苶 兹4
苶
4
(2 1)兹4
苶
4
3兹4
苶
Checkpoint Write the expression in simplest form.
3.
冪莦
4
4
2
9
4
18
兹苶
3
Example 5
4
4. 兹64
苶 兹4
苶
4
兹4
苶
Simplify Expressions Involving Variables
Simplify the expression. Assume all variables are positive.
4
81x12 a. 兹苶
4
兹苶
34(x3)4苶 3x3
b. (8a9b3)1/3 81/3(a9)1/3(b3)1/3 2a(9 p 1/3)b(3 p 1/3) 2a3b
c.
冪莦
6
6
6
兹苶
兹苶
y12
y12
(y2)6
y2
6 24
6
x24
x4
x
兹苶
兹苶
(x 4)6
8x2y6z2
d. 2x(2 1/4)y(6 5)z2 2x7/4yz2
4x1/4y5
Lesson 7.2 • Algebra 2 Notetaking Guide
157
Your Notes
Writing Variable Expressions in Simplest Form
Example 6
Write the expression in simplest form. Assume all variables
are positive.
冪莦 冪莦
冪莦
3
y2
x5
3
y2x
x5x
Make the denominator a perfect cube.
3
y2x
x6
Simplify.
3
兹苶
y 2x
3
兹苶
x6
Quotient property
3
2x
兹y苶
x2
Simplify.
Adding and Subtracting Variable Expressions
Example 7
Perform the indicated operation. Assume all variables are
positive.
3
3
3
3
苶 2兹x
苶
a. 3兹x苶 兹x苶 (3 1)兹x
b. 5x2y1/2 7x2y1/2 (5 7)x2y1/2 12x2y1/2
Checkpoint Simplify the expression. Assume all variables
are positive.
14xy3z5
5. 2x2/3yz2
3
3
6. 4兹苶
5y4 y兹40y
苶
Homework
7x1/3y 2z 3
158
Algebra 2 Notetaking Guide • Chapter 7
3
2y兹5y
苶