Unit 4 notes

Unit 4 Notebook 2012.notebook December 04, 2012

Simplifying Radicals
Irrational
Expressions

Nov 59:17 AM Nov 59:17 AM

Do you know your perfect squares???
List the first 15 perfect squares...
Did you know...
1 81
4 100
9 121
16 144
25 169
36 196
49 225
64


1 1
4 4
Can you simplify the following rational expressions? 9
16
What if the expressions were IRRATIONAL? Do you 9
16
Explain why each is rational! 25
36
know how to simplify these? 25
36
49 49
64 64
81 81
100 100
121 121
144 144
169 169
196 196
225 225
... ...


1


Steps to Simplifying Radicals: How would you simplify a radical that has a
variable inside of it???

1. Rewrite the irrational expression as the
Can you simplify the following:
product of 2 radicals. One of those radicals
MUST have a perfect square inside it.

2. Break down the radical with the perfect
square inside it.

3. Write your final answer.


Can you simplify the following irrational expressions? Can you simplify the following irrational expressions?
1
1
4
4
9
9
16
16
25
25
36
36
49
49
64
64
81
81
100
100
121
121
144
144
169
169
196
196
225
225
...
...


#3 #7

Homework:
p. 1‐2 #1‐ 21 odds
#11 #15


2


Answers to p. 1‐ 2 #1‐ 21 odds
Rationalizing Radicals
1. 3.

5. 7.

9. 11.

13. 15.

17. 19.

21.


1
1
So we know how to simplify ... 4
9
Rationalizing a denominator: 4
16 Removing all radical symbols from the 9
16
25
36 denominator of a fraction. 25
49 36
64 49
81
100
Does anyone know the best way to get rid of the 64
81
121 radical symbol in our example? 100
Do you know WHY we have to simplify 144 121
169 144
196 169
225 196
... 225
...


1

Rationalizing each expression below: 4
9
16
25
Steps to RATIONALIZING a fraction: 36
49
1. Simplify the expression if possible by dividing. 64

2. Multiply the top and bottom by the radical in 81
100

the denominator. 121
144
3. Simplify the remaining fraction. 169
196
225
...


3


1
Rationalizing each expression below: 4
9
16
25
36
49
64
81
100
121
Homework:
144
169
196
225
...
p. 3 #1- 8


#2 #4 Answers to p. 3 #1- 8

1. 2.

3. 4.
#6 #8
5. 6.

7. 8.


Do you recall how to simplify the following:
Adding & Subtracting Radicals
1. 2x + 3x

2. -5xy - 8xy

3. 7x2y - 17x2y + 9xy2


4


Then you should be able to simplify the
When you add or subtract radicals, you
following:
must have the same # inside the radical
1. symbol (radicand).
1
4
It's just like combining like terms!!! 9
16
25
2. 36
49
Ex: 64
81
100
121
144
3. 169
196
225
...


1

Use a bit of logic to do these problems. Simplify Simplify: 4
9

the radical that is easier to YOU first! Then you
1. 2.
16
25
know what the "last name" has to be for the other 36
49
radical. 64
81
100
121
144
Ex: 169

3. 4.
196
225
Which radical is easier to break ...
down?

What is the "last name" going to be?

5. 6.


#9 #15

Homework:
p. 4‐ 5 # 7‐ 27 odds #17 #21


5


Answers to p. 4 ‐5, #7‐ 27 odds QUIZ #6 TODAY 1
4
9
If you can simplify the following problems, then you should be ok ;) 16
25
36
7. 9. 11. 1. 2. 3. 4. 49
64
81
100
121

13. 15. 17. 144
169
196
225
...

19. 21. 23.

25. 27.


QUIZ #6 Do NOT do #2, 4 & 7
Journal Entry- Friday, Nov. 30
1
4
9
16
25
CR #6 due tomorrow Would you like to write a short paper on a
36
49
mathematician and create a poster based on their
64 "findings" to count as a test?
81
100
BONUS (3 points)
121
144
169
Simplify: Would you like to work in groups (3 people
196
225
max) or by yourself?
...

HAPPY FRIDAY!


Steps to Multiplying Radicals:
Multiplying
1. Multiply OUTSIDES with OUTSIDES
Radicals 2. Multiply INSIDES with INSIDES.
3. Simplify if possible.
Wh
at #
's a
re o
n th
Example: Wh
at 3
's a
re o
eO
UTS
IDE
n th ?
e IN
SID
E?


6


Try these. Multiply and simplify: 1 Uh oh‐ these have variables! Multiply and 1
4
9 simplify: 4
9
16 16
25
1. 2. 36
25
36
49
64
1. 2. 49
64
81 81
100 100
121 121
144 144
169 169
196 196
3. 4. 225
...
225

3. 4. ...


Using the Distributive Property: 1
4
(Remember, OUTSIDES with OUTSIDES, INSIDES with INSIDES)
9
16

1. 2. 25
36
49
64
81
Homework:
p. 6 # 1‐ 14
100
121
144
169
196
225
3. 4. ...


p. 6 #1 14
#3 #6 Answers to p. 6 # 1‐ 14

1. 2. 3.

4. 5. 6.

#9 #12 7. 8. 9.

10. 11. 12.

13. 14.


7


FOIL-ing Radicals Do you recall how to multiply binomials?
You guys call this the FOIL-ing Method.

Multiply and simplify:
1. (x + 2) (x - 9) =

2. (3 - x) (3 + x) =

3. (x2 - 1) (x2 - 7) =


Can you apply the FOIL-ing Method to Multiply and simplify:
RADICALS??? 1. 2.
Remember OUTSIDES with OUTSIDES and INSIDES with INSIDES!
1
1 4
4 9
9 16
Multiply and simplify: 16
25
25
36

1. = 36
49
49
64
64 81
81 100
100
121
3. 121
144
144 169
169 196
196 225
225 ...
...
2. =


Answers to p. 7 #21 28

21. 22.
Homework: 23. 24.
p. 7 #21 28 25. 26.

27. 28.


8


Activity? Pythagorean Theorem


y
10
9
8
C Mrs. E can't find her car in the parking lot
7
6 after a long trip at Target. Mrs. E is Mrs. E can't find her car in the parking lot
5
4
standing at T(-4, 2). After much ado, she after a long trip at Target. Mrs. E is
T 3
2
sees her car is parked at C (4, 8). Can you standing at T(-4, 2). After much ado, she
1 x determine how far away from her car she sees her car is parked at C (4, 8). Can you
10 8 6 4 2 0 2 4 6 8 10 is??? determine how far away from her car she
2
3 is???
4
5
6
7
8
9
10


Pythagoras (569-500 B.C.E.) was born on the island of Samos in Greece, and
did much traveling through Egypt, learning, among other things, mathematics.
DID YOU KNOW???
Not much more is known of his early years. Pythagoras gained his famous status
The DISTANCE formula was "created" by
by founding a group, the Brotherhood of Pythagoreans, which was devoted to
manipulating the Pythagorean Theorem.
the study of mathematics. The group was almost cult-like in that it had symbols,
rituals and prayers. In addition, Pythagoras believed that "Number rules the
universe,"and the Pythagoreans gave numerical values to many objects and ideas.
These numerical values, in turn, were endowed with mystical and spiritual
qualities.


9


Legend has it that upon completion of his famous theorem,
If we take an isosceles right triangle with legs of measure 1, the hypotenuse will
Pythagoras sacrificed 100 oxen. Although he is credited with the discovery of measure sqrt 2. But this number cannot be expressed as a length that can be
the famous theorem, it is not possible to tell if Pythagoras is the actual author. measured with a ruler divided into fractional parts, and that deeply disturbed the
The Pythagoreans wrote many geometric proofs, but it is difficult to ascertain Pythagoreans, who believed that "All is number." They called these numbers
who proved what, as the group wanted to keep their findings secret. "alogon," which means "unutterable." So shocked were the Pythagoreans by these
Unfortunately, this vow of secrecy prevented an important mathematical idea numbers, they put to death a member who dared to mention their
from being made public. The Pythagoreans had discovered irrational numbers! existence to the public. It would be 200 years later that the Greek
mathematician Eudoxus developed a way to deal with these unutterable
numbers...

1

1


The Pythagorean Theorem is used
to find the length of a side of a
RIGHT TRIANGLE.

a 2 + b 2 = c2

a & b represent the length of the
legs The Pythago
rean Theore
m also
states:
c represents the length of the
hypotenuse the area of
plus the are square A
is equal to th a of square B
***The hypotenuse is the longest side of the e area of sq
right triangle. uare C.


Pythagorean Triples:
Common lengths of sides in a right triangle. How do you tell if any given 3 sides
I bet you know at least one of these triples.... would form a right triangle? Use the
Pythagorean Theorem to check!!!
(3,4,5) (16, 63, 65)
Determine if the following lengths form
( 5, 12, 13) (20, 21, 29)
a right triangle.
( 7, 24, 25) (28, 45, 53)
1. 24 2. 15
( 8, 15, 17) (33, 56, 65) 10
14
( 9, 40, 41) (36, 77, 85)
26 7
(11, 60, 61) (39, 80, 89)

(12, 35, 37) (48, 55, 73)

(13, 84, 85) (65, 72, 97)


10


Find the missing side of the right triangle and round to the Find the missing side of the right triangle and leave your
nearest tenth. answer in simplest radical form.

2 6

12 15
3 4
2 10
6 1
5
5


Answers to p. 8 #1- 12

1. NO 2. YES 3. YES

Homework: 4. YES 5. NO 6. YES

p. 8 #1- 12 7. 8.9 8. 6.7

9. 12.2 10. 7.6

11. 7.3 12. 6.3


QUIZ #7 The Quadratic Formula
Ha ha
..
one is .This
funny, pretty
CR #7 due right??
?

tomorrow


11


Solve for x in the following 2 problems: Often, the simplest way to solve "ax2 + bx + c = 0"
for the value of x is to factor the quadratic, set
1. x2 5x 36 = 0 each factor equal to zero, and then solve each
factor.

2. 2x2 + 2x = 12
But sometimes the quadratic is too messy, or it
doesn't factor at all, or you just don't feel like factoring.

While factoring may not always be successful,
the Quadratic Formula can always find the
solution.


For ax2 + bx + c = 0, the value of x is given by: There are some songs that will help you memorize the
Quadratic Formula.

I use the song "Pop Goes The Weasel."
http://www.youtube.com/watch?v=2lbABbfU6Zc

This is a cute little video that students at Westerville South High School in Ohio created.
Maybe you guys could make one???
This formula is called The Quadratic Formula. (Start at 3:00)
http://www.youtube.com/watch?v=jGJrH49Z2ZA&feature=related
***Remember, a, b, and c represent the coefficients in our equation.
Solving ax2 + bx + c = 0 for x means, among other things, that you are trying to find xintercepts. This one is pretty bad...
http://www.youtube.com/watch?v=TVIcjaKt_A8&feature=fvwrel
Key Point: Sometimes they ask you to find the ROOTS.
This is a fancy way of asking you to solve for the variable!


Solve for x in each of the following in simplest radical Find the roots in each of the following equations in
form: simplest radical form:


12


ANSWERS to p. 10 11 #6, 8, and 16:

6.  x = 4,
Homework:
p. 10, 11 #6, 8, and 16
8.  x =

16.  n =


Quadratic Formula with Remind me again how you would solve the following quadratic
equation for the value of x in simplest radical form:

Sum and Product Rules 1.

Since this is an equation, there must be a way to check it. We could
substitute our answer back into the original equation, but our answer is really
messy. Plus there are 2 of them! There must be another method of check our
roots of the quadratic equation.


The method we will use to check our answers to a quadratic
equation is called the
SUM RULE
and
PRODUCT RULE

Product Rule:

Sum Rule:


13


Review


14

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