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7.2: Properties of Rational Exponents
[Algebra 2 (Y)]
HCPS III:
• Standard 10: Patterns, Functions, and Algebra: PATTERNS AND SYMBOLIC
REPRESENTATION: Use symbolic forms to represent, model, and analyze mathematical
situations.
• Benchmark MA.AII.10.8: Add, subtract, multiply, divide, and simplify rational
expressions, radical expressions containing positive rational numbers, and expressions
containing rational exponents.
Goal: Use properties of radicals and rational exponents.
Using Properties of Radicals
Product and Quotient Properties of Radicals
Property
Algebra
Product
Property
n
Quotient
Property
a•b = n a • n b
n
a na
=n
b
b
!
Example 1: Use Properties of Radicals
Simplify the expression.
!
3
3
3
3•3 9
a.) 16 • 4
b.)
!
5
c.)
!
48
4
3
d.)
!
SIMPLIFYING RADICALS
A radical with index n is in simplest form if there are:
• No perfect nth powers in the radicand
• No radicals in any denominators
Example 2: Write Radicals in Simplest Form
Simplify the expression.
a.)
3
104
!
!
4
96
5
3
b.)
3
d.)
4
40
!
c.)
3
1
32
!
1
8
Properties of Rational Exponents
Property
Algebra
Example
Product of Powers
a m • a n = a m +n
31 2 • 33 2 = 3(1 2+3 2) = 32 = 9
Power of a Power
(am ) = am•n
Power of a!Product
(ab)
Quotient of!Powers
!
Power of a Quotient
!
Negative Exponent
!
m
=!a m b m
am
m"n
=
a
!
an
" a % m! a m
$ ' = m
#b&
b
1
a"m = !
am
!
1
n
=a
"n
a
2
(23 2 ) = 2(3 2•2) = 23 = 8
12
(9 • 4 )
= 91 2 • 41 2 = 3• 2 = 6
53 2
1
( 3 2"1 2)
=
5
=
5
=5
12
5
" 8 %1 3 81 3 2
$ ' = 13 =
# 27 &
27
3
16"1 2 =
1
1
=
161 2 4
!
Example 3: Use Properties of !
Rational Exponents
Simplify the Expression.
!4
34
1
14 4
a.) 7 • 7
b.) (6 )
!
!
n
c.)
12
(49 •16)
!
!
13
d.) 64
83 2
e.) 81 2
f.)
!
75 2
71 2
!
Like Radicals
Like radicals have the SAME INDEX and SAME RAD ICAND .
e.g.,
3
2 and 4 3 2
Example 4: Add or Subtract Like Radicals
Simplify the expression.
!
!
5
5
5
5
7
12
"
12
a.)
b.) 4 3 + 3
!
!
c.)
!
4 (9 2 3 ) + 8(9 2 3 )
13
13
7
11
"10
11
) ( )
d.) (
!
Simplifying Variable Expressions
Example 5: Simplify Expressions with Variables
Simplify the expression. Write your answer using positive exponents only.
Assume all variables are positive.
16x 4
a.)
!
!
8 14
c.) (16y )
!
!
9x 6
b.)
6 12
d.) (4 y )
!
e.)
3
x6
y9
f.)
!
3
x3
y6
4 y5 4z
g.) yz"3
!
!
Example 6: Add and Subtract Expressions with Variables
Simplify the expression. Assume all variables are positive.
b.) 4 x " 3 x
a.) 5 y " 2 y
!
!
!
3a 3 2c
h.) ac "2
2 34
2 34
c.) 6x y + 3x y
14
14
d.) 5xy + 2xy
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