Chapter 7 – Radical Functions and Rational

Exponents18 Days

Roots and Radical Expressions

Four Days

Review of Exponent Rules

Definition of the nth root:

What is a root??

. ofroot th an is then

, if ,integer positiveany and , and numbers realany For

bna

banba n

A few examples..

16 of roots4th are 2- and 2 16)2( and 16)2(

8 ofroot 3rd theis 2 8)2(44

3

Terminology for Radicals

(n) root theof degree the-Index

(a)root under thenumber the- Radicand

n a

Lets start with a few familiar examples:

Simplifying Radicals

23

4

2

100

36

4

24

16

yx

x

x

Now lets increase the index

3 463

4 6

2

3 4

3 6

64

81

25.

16

27

zyx

x

x

x

x

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

 x2                              

 x3                              

 x4                              

 x5                              

Powers of Real Numbers

pg 372 (# 1-27 odd)

Homework

7.2 Multiplying and Dividing Radicals

Three Days

Warm-up

33 5416

3612552

2428

For a Radical Expression to be in simplest for the following conditions must be met:

◦ No perfect nth power factors, other than 1.

◦ No fractions in the radicand.

◦ No radicals in the denominator.

Simplest form of a Radical

Multiplying and Dividing Radicals

nn baba nnn then numbers, real are b and a If

nn b

a

b

a

nnn then numbers, real are b and a If

Lets try a few examples..

4 64 3

3 53 2

33

327

164

93

123

xyyx

xyyx

Lets try a few examples..

x

x

yx

3

12

8

27

8

216

36

47

4

3

3 36

3

3

Rationalizing the denominator of an expression is the process of re-writing so that there are no radicals in any denominator and there are no denominators in any radical.

Rationalizing the Denominator

Lets rationalize the denominators

3

3

3

6

4

5

3

1

3

2

x

xy

x

x

pg 377 (# 1-35 odd)

Homework

Practice 7.2 WS (1-33 odd, 34)

Homework

Practice 6-5 (#1-35 odd) - Glencoe

Homework

7.3 Binomial Radical Expressions

Three Days

7.4 Rational ExponentsThree Days

Review of Exponent Rules

First of all, what is a rational number? It’s a number that can be written as a

fraction of integer values.

What are Rational Exponents?

mnn m

n

xxx

xx

nma

nm

n

and

thenintegers, are and andnumber real a is ofroot nth theIf

Exponents Rational of Definition

1

Re-writing expressions

3 4

5

5.3

23

z

a

y

x

Simplifying Rational Expressions

31

145

72

45

6

5.2

27

4

16

x

xx

pg 388 (# 1-25 odd, 39-49 odd)

Homework

7.5 Solving Radical Equations

Three Days

Solve the following radical expression:

Warm-up

6422 x

What steps did you follow to solve the radical?

Here is another representation..

6422 21

x6422 x

1. Convert the radical to rational exponents.

2. Isolate the “radical” part of the equation.

3. Raise both sides of the equation to the reciprocal power of the rational exponent.

4. You’ve now “cleared” the radical, solve using the appropriate method for the resulting equation.

Lets develop some steps to solve radicals equations..

Lets try a few..

22422 32

x

Lets try a few..

xx 57

The maximum flow of water in a pipe is modeled by the formula Q=Av, where A is the cross sectional area of the pipe, v is the velocity of the water, and Q is the maximum volume of water than can flow through the pipe per minute.

Find the diameter of a pipe that allows a maximum flow of 50 cubic feet per minute at a velocity of 600ft/min. Round to the nearest inch.

Applications

pg 394 (# 1-25 odd)

Homework

Practice 7.5 WS (#2-32 Even)

Homework

Practice 6-7 WS (# 2-22 even) - Glencoe

Homework

7.8 Graphing RadicalsThree Days

Parent: Shift up k units: Shift down k units:

Shift right h units: Shift left h units

Combined Shift:◦ (right h units, up k units)

Graph Shifting and Reflections

xy

kxy

kxy

hxy

hxy

khxy

)(xfy

khxfy )(

)( hxfy

)( hxfy

kxfy )(

kxfy )(

Parent: Reflection in x-axis:

Vertical Stretch a>1 Vertical Shrink 0<a<1

Horizontal Stretch 0<c<1 : Horizontal Compression c>1:

Combined Transformation:

Graph Shifting and Reflections

xy

xay

xcy

xay

)(xfy

)(xfay

)( xcfy

)(xfay

khxay khxfay )(

-Name of Family

-Parent Equation

-General Equation

-Locator Point

-Domain -Range

Parent Functions

Root Square

xy

khxay

),(:Endpoint kh

),0[ ),0[

8

6

4

2

-2

-4

-6

-8

-15 -10 -5 5 10 15

f x = x

x y0 01 14 29 316 4

-Name of Family

-Parent Equation

-General Equation

-Locator Point

-Domain -Range

Parent Functions

Root Cube

3 xy

khxay 3

),(:Inflection kh

8

6

4

2

-2

-4

-6

-8

-15 -10 -5 5 10 15

f x = x1

3

x y-8 -2-1 -10 01 18 2

Graph the following

1

3

xy

xy

Graph the following

4

2

xy

xy

Graph the following

31 xy

Graph the follwing

24

4

3

3

3

xy

xy

xy

https://www.desmos.com/calculator/renedj48tv

Interactive Graph of Radicals

pg 417 (# 1-23 odd)

Homework

Practice 7.8 WS (# 1-15 odd, 28, 29, 31, 35, 37)

Homework

Homework