Please
CLOSE
YOUR LAPTOPS,
and turn off and put away your
cell phones,
and get out your notetaking materials.
Math 110 Final Exam:
•
•
•
•
•
Comprehensive – covers the whole semester
Worth 200 points (20% of course grade)
45 questions
Test period is one hour and 50 minutes
Practice Final will be worth 10 points and also has 45 questions
and unlimited tries.
• Your best score on the practice final will also earn up to 20 extra
credit points on the final. (more details next week)
Make sure you know the
day and time of the final exam
for this section of Math 110:
• All Math 110 finals will be given in
your regular classroom.
• (Next slide shows final exam
schedules for all sections.)
Wednesday 12/16
Thursday 12/17
Friday 12/18
Monday 12/21
Scheduled Final in 214:
110-001 Neiderhauser 8:00
Scheduled Final in 214:
Scheduled Final in 214:
Scheduled Final in 214:
110-002 Lee 10:10
010-003 DKL MW 12:20
110-005 Lee 2:30
Tuesday
12/22
8:00
to
10:00
Scheduled Final in 245:
010-004 Corne TTh 2:30
10:00
to
12:00
Scheduled Final in 214:
Scheduled Final in 214:
Scheduled Final in 214:
Scheduled Final in 214:
110-006 Lee 3:35
010-001 DKL MW 9:05
110-003 Schmidt 11:15
110-004 Thielman 1:25
12:00
to
2:00
LAB CLOSED
Scheduled Final in 214:
010-002 Mayer TTh 9:05
2:00
to
4:00
Scheduled Final in 214:
110-007 Hulett 4:40
LAB CLOSED
4:00
to
6:00
Scheduled Final in 214:
110-008 Hulett 5:45
6:00 to
8:00
LAB CLOSED
LAB CLOSED
Section 10.2
Radicals and Rational Exponents
Definition of a rational exponent in
terms of a radical:
If n is a positive integer greater than 1 and
a is a real number, then
1/ n
a
 a
n
Why does this definition make sense?
Recall that a cube root is defined so that
3
a  b only if b 3  a
However, if we let b = a1/3, then
b3  (a1/ 3 )3  a 1/ 33  a1  a
Since both values of b give us the same a,
a1/ 3  3 a
Example
Use radical notation to write the following.
Simplify if possible.
811/ 4 
32x 
10 1 / 5
4
81  4 34  3
 5 32 x10  5 25 x10  2 x 2
We can expand our use of rational exponents
to include fractions of the type m/n, where
m and n are both integers, n is positive, and
a is a positive number,
a
m/n
 a 
n
m
 a
n
m
Example
Use radical notation to write the following.
Simplify if possible.
8
4/3

 8
3
4

2
3
3
4
24 
16
Problem from today’s homework:
64
Now to complete our definitions, we want to
include negative rational exponents.
If a-m/n is a nonzero real number,
a
m / n

1
a
m/n
Example
Use radical notation to write the following.
Simplify if possible.
64
 16
2 / 3
5 / 4
1


2/3
64
1
 64 
2
3
1
  5/ 4  
16

2
4
4
3
1
4
1
5
3
2
1
1
 2 
16
4
1
1
  5 
2
32
5 / 4
(

16
)
What if the previous problem was
?
The answer would be “N” (not a real number) because you’d be trying to take an
even root of a negative number.
All the properties that we have previously
derived for integer exponents hold for rational
number exponents, as well.
We can use these properties to simplify
expressions with rational exponents.
Example
Use properties of exponents to simplify the
following. Write results with only positive
exponents.
32
1/ 5
x

2/3 3
 32
3/ 5
x 
2
 2  x
5
5
3
2
3
2
2
2

x

8x

a1/ 4  a 1/ 2
1

1 / 4  1 / 2   2 / 3 

3 / 12 6 / 12 8 / 12 
11/ 12
 a
a
a
 11/12
2/3
a
a
𝟏
What would this answer look like in radical form?
𝟏𝟐
𝒂𝟏𝟏
Problem from today’s homework:
Final answer: -b
Hint: The exponent will be 2/5 + 1/5 – (-2/5).
Then simplify the fraction.
Example
Use rational exponents to write as a single
radical.
3
5 2  5 2
1/ 3
1/ 2
 5
2/6
2
3/ 6

 5 2
2

3 1/ 6

6
200
Problem from today’s homework:
Final answer:
20
y19
Hints: Start by writing each radical as a rational (fraction) exponent,
then add the fractions by finding a common denominator.
For your final step, convert back into radical form.
The assignment on this material (HW 10.2)
is due at the start of the next class session.
You may now OPEN
your LAPTOPS
and begin working on the
homework assignment until
the end of the class period.