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