### Sec. 10.2: Radicals and rational exponents

```Please open Daily Quiz 34.
• A scientific calculator may be used on this quiz.
• You can keep your yellow formula sheets out
when you take the quiz.
when the quiz time is up.
If you have any time left after
finishing the quiz problems,
before you submit the quiz.
CLOSE
and turn off and put away your
cell phones,
and get out your notetaking materials.
Last two weeks of class:
• Today: (Monday, 12/2)
• Daily Quiz 34 on HW 39 due today on Section 10.1
• Hand back/review Quiz 9
• Lecture on Section 7.1/2 (Rational exponents)
• Tuesday –Thursday this week:
• Daily Quizzes on HW due each day (DQ 35, 36, 37)
• Lectures on last 3 new sections (10.3&4, 11.2, 11.5)
•
NEXT WEEK: Monday, 12/9:
• Practice Weekly Quiz 10 due at start of class
• 10-15 minutes of Q&A and review
• 25-point Weekly Quiz 10 on all sections since Test 3
• Tuesday –Thursday next week:
• Review lectures on Units 1-3 (Tests 1-3)
• Review HW on each unit, worth double points (8 pts each)
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:
Day: ______ Date:______
Time: ______ to _______
• All Math 110 finals will be given in your regular
classroom.
• (Next slide shows final exam schedules for all
sections.)
Weekly Quiz 9 Results:
•Average class score after partial credit: __________
•Commonly missed questions: #_________________
A
A-
B+
B
B-
C+
C
C-
F
Points
≥ 920 ≥ 890 ≥ 860
≥ 830
≥ 800
≥ 750
≥ 700 ≥ 670 < 670
% Score
≥ 92
≥ 83
≥ 80
≥ 75
≥ 70
≥ 89
≥ 86
≥ 67
< 67
If you got less than 75% on this quiz, make sure to go over your
test with me or a TA sometime in the next few days. This material
will be covered on the next weekly quiz, and it will also be covered
again on the final exam.
Section 10.2
Definition of a rational exponent in
If n is a positive integer greater than 1 and
a is a real number, then
a
1/ n

n
a
Why does this definition make sense?
Recall that a cube root is defined so that
3
a  b only if b  a
3
However, if we let b = a1/3, then
b  (a
3
1/ 3
) a
3
1 / 3 3
a a
1
Since both values of b give us the same a,
a
1/ 3

3
a
Example
Use radical notation to write the following.
Simplify if possible.
81
1/ 4
32 x 
10 1 / 5

4

5
81 
32 x
4
10
3 3
4

5
5
2 x
10
 2x
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

n
a
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
64
 
2/3
1
16
5/4


3
 
1

2
64

2
4
4
3
1
4
1
5
 
What if the previous problem was
2
3
1
2
5
1

4
 
(  16 )
2

1
16
1
32
5 / 4
?
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
a
1/ 4
a
a
1/ 5
x
1 / 2
2/3

2/3 3
 a
 32
3/5
x 
1 / 4  1 / 2   2 / 3 
2
 a
 2  x
5
5
3
2
 23  x 2  8 x 2
 3 / 12   6 / 12   8 / 12 
 a
 11 / 12
1

a

11 / 12
Problem from today’s homework:
Hint: The exponent will be 2/5 + 1/5 – (-2/5).
Then simplify the fraction.
Example
Use rational exponents to write as a single
3
5 2  5
1/ 3
2
1/ 2
 5
2/6
2
3/6

 5 2
2

3 1/ 6

6
200
Problem from today’s homework:
20
y
19
Hints: Start by writing each radical as a rational (fraction) exponent,
then add the fractions by finding a common denominator.