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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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