Please
CLOSE
YOUR LAPTOPS,
and turn off and put away your
cell phones,
and get out your notetaking materials.
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 10.3
Simplifying Radical Expressions
Recall these square root problems from Section 10.1:
Examples:
72 = (72)½ = 71 = 7
49 =
24 = 2 2 = 4
16 =
24 ∙ 52 = 22 ∙ 5 = 20
400 =
𝑥 6 𝑦10 =
25𝑎2 𝑏14
=
𝑥 3𝑦5
52 𝑎2 𝑏14 = 5𝑎𝑏 7
What we did in the previous examples was essentially
to divide the exponent of each base by 2,
which is index of the radical for square roots.
But what happens if the radicand ( the expression under
the radical) is not a perfect square, i.e. has exponents that
are not divisible by 2?
Example: How would we simplify 𝒙𝟕 ?
Solution:
𝒙𝟕 =
𝑥
7/2
= 𝑥
3½
3 ½
=𝑥 𝑥
3
𝑥
𝑥
=
• Think of this as dividing the exponent 7 by the index 2
• Two goes into seven 3 times with a remainder of 1
𝑥 3 𝑥1
Example
Simplify
9
12 x y
12
First, break down 12 into its prime factors:
12 = 4∙3 = 2∙2∙3 = 22∙31
This gives
9
x y
12
Now divide the exponents by 2 (square root, so the index is 2).
Answer:
4
2x y
6
3x
Problem from today’s homework:
-
2232111x6y15
Final Answer:
 6 x 3 y 7 11 y
Start by breaking down 396:
(use your calculator, and start by dividing by 2)
So 396 = 2232111
396
2 198
2
99
9 11
3 3
If we have a radical with an index of 3 or higher, we
can use the same process to simplify the radical.
𝟑
Example: How would we simplify 𝒙𝟕 ?
Solution:
• Divide the exponent 7 by the index 3
• Three goes into seven 2 times with a remainder of 1
3
𝑥7
=𝑥
2 3
𝑥1
Example
Simplify
3
2
10 6
40 x y z
Answer:
2 y3 z 2 3 5x2 y
Problem from today’s homework:
8 17 4 7
qr s
3 5
qr
Product and Quotient Rules for Radicals:
If n a and n b are real numbers, then
Product Rule:
n
a  n b  n ab
Why is this
n
condition
Quotient Rule: a  n a if b  0
n
important?
b
b
Because division by zero gives an undefined quotient.
Does 3  3  (3)   3  9  3 ?
No, because square roots of negative numbers
are not real numbers.
Example Simplify the following radical expressions.
(Assume x and y are ≥ 0)
3 y  5x 
15 xy
(Assume a > 0 and b ≠ 0)
7 6
ab
3 2
ab

7 6
ab

3 2
ab
ab  ab
4 4
2 2
Problem from today’s homework:
3
Problem from today’s homework:
5
Example
Use the quotient rule to divide, then simplify if
possible:
83 54m 7
3
2m
Answer:
24m 2
• In previous chapters, we’ve discussed the
concept of “like” terms.
• These are terms with the same variables
raised to the same powers.
• They can be combined through addition and
subtraction.
Example: (x2 + 5x – 1) + (6x2 - 3x + 4) =
7x2 + 2x + 3
• Similarly, we can work with the concept of
“like” radicals to combine radicals with the
same radicand.
• Like radicals are radicals with the same
•
index and the same radicand.
Like radicals can also be combined with
addition or subtraction by using the
distributive property.
Example
37 3  8 3
10 2  4 2  6 2
2 4 2
3
5 3
Can not simplify (different indices)
Can not simplify (different radicands)
Always simplify radicals FIRST to determine
whether there are like radicals to be
combined.
Example
Simplify the following radical expression.
 75  12  3 3 
 25  3  4  3  3 3 
 25  3  4  3  3 3 
5 3  2 3 3 3 
 5  2  3
3  6 3
REMINDER:
The assignment on today’s material (HW 10.3) 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