Honors Advanced Algebra
with Trigonometry
Section----7.6 Notes
#5 (p. 446)
Rational Exponents
Opener Solutions:
1. 7 2 + 4 18 " 50 = 14 2
!
2. 3192t " 3 24t +3 !5t = 23 3t + 3 5t
!
3. #$% 3 5 " 3 &'(#$% 3 5 !+ 3&'( = 42
!
4. #$%5 6 + 3&'( #$%2 6 " 4 3&'( = 48 " 54 2
!
5. 4 3 " 7 = 102 "3959 ! 3
!
Opener Solutions:
3
3
6. 4x 5 = 2 180x2
3 6x
3
7. Find f(g(x)) if f(x) = x2 + 2x
!
!
and g(x) = x - 3.
x2 - 4x + 3
5 3 +6
!
Target Goals:
• Write expressions with
rational exponents in
simplest radical form and
vise versa
• Simplify expressions in either
exponential or radical form.
Notes:
Example One:
1
D: bn = n b where n is a
positive integer.
1
ex. 3 8 =83 =2
Example One:
b. Write 6 x5 in exponential form.
5
x6
Note: The 5!becomes numerator of the
exponent, the 6 becomes the denominator of the
exponent and the x is the base.
2
a. Write x 3 in radical form.
3 x2
Note: The 3 becomes the index, the x is the
radicand, and the 2 is the exponent.
Example Two:
1
a. Evaluate 814
= 4 81 = 3
calculator: 81^(1/4) enter
or rewrite as: 4 34 = 3
Example Two:
b. Evaluate
#
%
%
%$
D: Rational Exponents
"1
1 3
8
m
n =n m
b
b
&
(
(
('
calculator: (1/8 )^("1/3) enter
or rewrite as:
#
%
%
%$
1
8 3 = 3 23
1
&
(
(
('
=2
Example 3:
Express each in radical form
!need common denominator"
1
3
1
2
a. 4 a b
5
6
1
2
3 1
4 5
b. x x x
4
ex. 8 3= 3 84 =16
On Calculator: 8^(4/3) enter
1
3
1
2
2
6
3 5
6 6
3a. 4 a b
5
6
=4 a b
Need a common denominator
for the exponents.
2
6
3 5
6 6
=4 a b
1
3 5 6
=( 4 a b
2
)
1
3 5 6
=(4 a b
2
)
= 6 16a3b5
Rewrite in radical form.
Move the 16 outside
parentheses as an exponent.
!
1
2
3 1
4 5
3b. x x x
Try with your group.
1
2
3 1
4 5
3b. x x x
= x
10 15 4
+ +
20 20 20
=
20
=x
29
20
x 29
Still needs to be simplified.
=
20
= x
Example 4:
20 9
x x
20
x
Simplify.
Leave solutions in simplest radical form.
"5
"4 3
b. m 5
#
%
%
%
%
%
%
%
%
%
$
7
a. b 6
9
!
&
(
(
(
(
(
(
(
(
(
'
!
4b.
4a.
"5
4
" 3
4
"4 "5
5
5
3
m
=m
= m3
#
%
%
%
%
%
%
%
%
%
$
7
6 1
6
b = b6 )b6 = b 6 b
&
(
(
(
(
(
(
(
(
(
'
#
%
%
%
%
%
%
%
$
&#
(%
(%
(%
(%
(%
(%
(%
'$
!
!
Not done yet . . .
&
(
(
(
(
(
(
(
'
4b.
Example 5
4
3 1
3
= m = m3)m3 = m3 m
Work with your group.
!
Example 5
a.
5 11
4
x x3
b.
=x4 12 x11
Example 6:
#
%
%
%
%
%
%
%
%
$
"7
2 4
z7
&
(
(
(
(
(
(
(
(
'
= zz
!
!
Simplify. Leave solutions exact, in exponential
form, with only positive exponents.
a.
83521
b.
!
!
3 81x11y5
3 15x8y7
Example 6:
a.
Example 6:
a.
83521
1
12
= 835212
#
%
%#
%%
%%
%%$
%
%
$
&
(
((
'
83521
1
12
= 835212
#
%
%#
%%
%%
%%$
%
%
$
&
(
(
(
(
(
(
(
'
&
(
((
'
&
(
(
(
(
(
(
(
'
1
= 83521 4
=17
#
%
%%
$
!
&
(
((
'
!
Example 6:
b. 3 81x11y5
3 15x8y7
Example 6:
b.
=3 27x3
5y2
=3 81x11y5
15x8y7
!
=3 81x11y5
15x8y7
!
Example 6:
b.
=3 27x3
5y2
= 3x
3 5y2
!
Example 6:
b. = 3x )3 52 y
3 5y2 3 2
5 y
Now
Rationalize
#
%
%%
$
3x 25y
3
= 3x 25y =
5y
5y
&
(
((
'
1
3
!
Example 7
Example 7
a.
625
1
12
= 625 2
Work with your group.
#
%
%#
%%
%%
%%$
%
%
$
&
(
((
'
1
=6254
=5
!
&
(
(
(
(
(
(
(
'
Example 7
4 32x5
= 4 32x5
4 16x6
16x6
b.
=
42 4 3 4 3
) x = 2x
x
4 x 4 x3
2x
= x
#
%
%
$
&
(
(
'
3
4
!
Homework
#5
Worksheet
Prepare for QUIZ
Covering HW #1-#4 and factoring
See board for date.
Target Goals:
• Write expressions with
rational exponents in
simplest radical form and
vise versa
• Simplify expressions in either
exponential or radical form.