Transcript 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.
Remember to turn in your answer sheet to the TA when the quiz time is up.
If you have any time left after finishing the quiz problems, CHECK YOUR ANSWERS before you submit the quiz.
Please CLOSE
YOUR LAPTOPS,
and turn off and put away your cell phones,
and get out your note taking 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
• 25-point
Weekly Quiz 10
due at start of class • 10-15 minutes of Q&A and review 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: #_________________
Grade A A-
Grade Scale
B+ B B- C+ C C- F ≥ 920 ≥ 890 ≥ 860 ≥ 830 ≥ 800 ≥ 750 ≥ 700 ≥ 670 < 670 Points % Score ≥ 92 ≥ 89 ≥ 86 ≥ 83 ≥ 80 ≥ 75 ≥ 70 ≥ 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
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
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
3
a
However, if we let b = a 1/3 , then
b
3 (
a
1 / 3 ) 3
a
3
a
1
a
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 4 81 4 3 4 3 32
x
10 1 / 5 5 32
x
10 5 2 5
x
10 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
n a m
n
m
Example
Use radical notation to write the following. Simplify if possible.
8 4 / 3 3 4 3 4 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 2 / 3 1 64 2 / 3 1 3 2 3 1 2 1 4 2 1 16 16 5 / 4 1 16 5 / 4 1 4 5 1 2 5 1 32 ( 16 ) 5 / 4 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 5 3
x
2 2 3
x
2 8
x
2
a
1 / 4
a
1 / 2
a
2 / 3
a
2 / 3
a
3 / 12 6 / 12 8 / 12
a
11 / 12 1
a
11 / 12 What would this answer look like in radical form?
𝟏 𝟏𝟐 𝒂 𝟏𝟏
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 1 / 3 2 1 / 2 5 2 / 6 2 3 / 6 5 2 2 3 1 / 6 6 200
Problem from today’s homework:
Final answer: 20
y
19 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 .
REMINDER:
The assignment on today’s material
(HW 40)
is due at the start of the next class session.
Please open your laptops and work on the homework assignment until the end of the class period.
Lab hours in 203: Mondays through Thursdays 8:00 a.m. to 7:30 p.m.
Please remember to sign in on the Math 110 clipboard by the front door of the lab