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Roots and Radical Expressions
52= so √_ =5
53= so 3√_ =5
54= so 4√_ =5
55= so 5√_ =5
What are some cubic roots we know?
Graph on a number line: ∛7, ∛23, ∛83
What are some 4th roots we know?
Graph on a number line: ∜7, ∜101
PRISON BREAK! Simplify each expression. Assume all variables are positive.
A. √(72x3) B. 3√(80n5)
C. √(50x4) D. 3√(18x4)
Write your own example with a 4th root. Write your own example with a 5th root.
Rationalizing the Denominator
We don't want a √ in the denominator of a fraction.
Examples
1
√2
7
√10
5
∛3
3
∛4
9
∜2
1
∜x
8
∜xy2
10a
∜bc2d3
Negative x Negative = (Negative)2 = _________
Negative x Negative x Negative = (Negative)3 = __________
(Negative)4 = __________
(Negative)5 = __________
(Negative)6 = __________
(Negative)7 = __________
(Negative)8 = __________
What is the pattern?
(-5)3= so 3√ =-5
(-5)5= so 5√ =-5
(-5)7= so 7√ =-5
However,
(-5)2=(5)2= , √ =
(-5)4=(5)4= , 4√ =
(-5)6=(5)6= , 6√ =
With a negative radicand, an ______________________has an _____________answer and an ______________________________ has a _____________________ answer.
What is the difference between the EXACT answer and APPROXIMATE answer to ∛-16 ?
Simplify. Assume all variables are positive.
A. 3√8 B. 3√-8
C. 5√3125 D. 3√-125
E. 4√(x4y8) = F. 3√(-27c6) =
G.3√(-27c7)=
What does x equal? Be careful. Some may have more than one answer.
x = 3
x2 = 9
x3 = 27
x4 = 81
x5 = 243
x6 = 729
Rule of Thumb #1
Give only positive answer if...
Ex:
Give both answers if...
OR
Ex: Ex:
Rule of Thumb #2
When you square root both sides of an equation you don't forget to write .
The same goes for.
Ex:
Find the real roots.
A. cubic roots of 27
B. cubic roots of -27
C. square roots of 16
D. fourth roots of 625
Solve
A. x3 = 27
B. x3 = -27
C. x2 = 16
D. x4 = 625
Simplify
A. ∛27
B. ∛-27
C. √16
D. ∜625
To _______________ or _______________ radical expressions you must have the same _______________.
To _______________ or _______________ radical expressions you must have the same _______________ AND the same _______________.(called _______________ _______________.)
Which ones can you do? (Circle)
2√7 * 3√5 2√7 + 3√5
2√7 6∛4 - 5∛10 6∛4 * 5∛10
3√5
6∛4 8∜11 * 9∜11 8∜11 + 9∜11
5∛10
Rule of Thumb #3
When you +, - , x, or ÷ whatever ________________________ will be _____________________________________________.
Ex:
To multiply or divide: Multiply the outsides together and the insides together.
A. √2 ∙ √8 =
B. 3√-5 ∙ 3√25 =
C. √-2 ∙ √ 8 =
D. √3 ∙ √12 =
E. 3√3 ∙ 3√-9 =
F. 4√4 ∙ 4√-4 =
G.3√(54x2y3) ∙ 3√(5x3y4) =
H.3√(7x3) ∙ 2√(21x3y2) =
I. √36
√25
J. ∛32
∛-4
K. ∛162x5
∛3x2
To add or subtract: Add the _____________ together.
A. 5∛x - 3∛x =
B. 2√7 + 3√7 =
C. 4√2 + 5√3 =
D. 7∛5 - 2∜5 =
E. 4√xy + 5√xy =
Simplify
A. 6√18 + 4√8 - 3√72 B. √50 + 3√32 - 5√18
C. 14√5 - 2√12 - 3√20
Multiply.
A. (3 + 2√5)(3 + 2√5)
B. (2 + √3)(2 - √3)
Rationalizing the Denominator
We don't want a √ in the denominator of a fraction.
Examples
3 + √5
1 - √5
6 + √15
4 - √15
Rational Exponents
Warm-up: Simplify
83(82)=
(52)4=
(5x)2=
9-2=
Since √7= ∛11= ∜13=
What do you suppose equals 191/5?
Example - Convert from radical form.
A. ∜x3= B. √a7= C. (∛b)2=
Example - Convert to radical form and simplify.
A. y-5/2= B. x-3/8= C. (-32)3/5=
D. 25-3/2= E. (-32)4/5=
Example - Decimal Exponents. Convert to Radical Form.
A. y-2.5= B. x1.5= C. z1.2=
Warm-up:
Convert from radical form to rational exponent form.
A. ∛a2 = B. (∛a)2=
Properties of Rational Exponents
aman=am+n Example: 81/3 x 82/3 =
(ab)n=anbn Example: (4x)1/2 =
(am)n=amn Example: (51/2)4 =
a-n= 1/an Example: 9-1/2 =
am/a n =am-n Example: [pic]
(a/b) m=am/bm Example: [pic]
Example: Write in Simplest Form.
A. (16y-8)-3/4 B. (8x15)-1/3 C. (x2/3y-1/6)-12
D. [pic] E. [pic]
Physics: In the expression PV7/5, P represents the pressure and V represents the volume of a sample of gas. Evaluate the expression for P=6 and V=32.
Solving Radical Equations
A ____________________ is an equation with a variable under a radical sign or a variable with a rational exponent.
Radical Equations NOT Radical Equations – why not?
To solve, we want to…
1. Get radical ___________________.
2. _________________ of radical.
3. If more than one radical ____________ steps 1 and 2.
4. _____________ equation.
Example – Solve 2 + [pic]= 6 .
Practice – Solve [pic] - 6=0 .
Example – Solve 2(x – 2)2/3 = 50
Examples: Solve. Check for Extraneous Solutions.
[pic]
[pic]
[pic]
Function Operations
Example 1 - A boutique prices merchandise by adding 80% to its cost.
It later decreases by 25% the price of items that don’t sell quickly.
• Write a function f(x) to represent the price after the 80% markup.
• Write a function g(x) to represent the price after the 25% markdown.
• Use a composition function to find the price of an item after both price adjustments that originally costs the boutique $150.
• Does the order in which the adjustments are applied make a difference? Explain.
Example 2 - Let f(x)=4x - 1 and g(x)=2x2 + 3. Perform each function operation. Find domain.
A. f(x) + g(x)
B. f(x) - g(x)
C. f(x) ∙ g(x)
D. f(x)
g(x)
E. g(x)
f(x)
F. f○g
G. g○f
Example 3 - Let f(x)=-3x + 2, g(x)=x/5, h(x)=-2x2 + 9, and j(x)=5 - x. Find each value.
A. (f○j)(3)
B. (j○h)(-1)
C. (h○g)(-5)
D. 3f(x) + 5g(x)
Inverses
How to Find the Inverse of a Function
1. ______________ the equation using y, if necessary.
2. ________________ x and y.
3. __________________for y.
4. Write f-1(x) in place of _____________.
Note: f-1(x) means:
Example: Find f-1(x) (the inverse)
f(x)=x+1
3 Nifty Things About Inverses
1. Inverses _________________.
Let f(x)=2x-8 Find the inverse.
Start with 10. (It will work with any number.)
Find fof-1(10)
Find f-1of(10)
2. Inverses interchange the _________________ and _______________.
3. Inverses ________________________ across the line _______________.
x |-1 |0 |1 |1 | |y |1 |2 |3 |4 | |
Graph the Relation.
Graph the line y=x.
x | | | | | |y | | | | | |
Graph the Inverse.
Practice graphing the inverses of the functions.
y=x2+3 y=3x-10
Graphing Radical Functions
[pic]
Note:
The graph passes through the points:
We can easily alter this basic equation and graph.
If we have no stretch/compress factor:
Move the "____________________" from ___________
Translate the other points that still fit on the graph ____________________________
Example 1 – Graph [pic]
Practice - Graph [pic] Practice - Graph [pic]
Stretch/Compress ___________________ translating. Just ______________ the ______ value of your points by a.
Next translate.
Example 2 - Graph [pic]
Practice - Graph [pic]
One Last Thing!
When you have a negative in front, ________ [pic]
below the x-axis before anything else.
Example 3 – Graph [pic] Practice – Graph [pic]
Practice – Graph [pic]
[pic]
Example 1: y = [pic] Example 2: y = [pic]
Practice: A. y = [pic] B. y = [pic]
Example 3: y = [pic] Practice C: y = [pic]
n Factorial
Note: 0! = 1! =
So
0!= 1!= 2!= 3!=
3!4!= 6(3!)=
10!= 2!/4!=
On the Calculator:
Arrow to PRB 4:!
Auto mechanics advise that tires on a car should be rotated every 6000 miles.
In how many ways can 4 tires be arranged on a car?
If the spare tire is included, how many arrangements are possible?
• How many different 9-player batting orders can be chosen from a team of 16 players?
• How many different groups of "starters" can be chosen from a soccer team of 16 (11 players on the field)?
Multiplication Counting Principle:
In how many different orders can 10 dogs line up to be groomed?
The prom committee has 4 sites available for the banquet and 3 sites for the dance. How many arrangements are possible?
A car door lock has a five-button keypad. How many different five digit patterns are possible. (Hint: You can use a button more than once.)
• Ten candidates are running for 3 seats in the student government. You may vote for as many as 3 candidates. In how many ways can you vote for 3 or fewer candidates?
• For a camp, you can choose 2 or 3 roommates from a group of 25 friends. In how many ways can you choose?
• A consumer magazine rates televisions by identifying 2 levels of price, 5 levels of repair frequency, 3 levels of features, and 2 levels of picture quality. How many different ratings are possible?
-----------------------
n
√a
Notice: It is OK to have a negative under an _______ index.
x3
x2
=
[pic]
Remember:
(a+b)n≠an+bn
1. Sketch __________________
2. ______________ points vertically.
3. _______________________
n!=
MATH
Note: You can find more info about this section in ch6 of your Algebra II book.
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