Transcript Chapter 7 Powerpoint

Chapter 7
Radical Functions
and Rational
Exponents
In this chapter, you will …



You will extend your knowledge of roots to
include cube roots, fourth roots, fifth roots,
and so on.
You will learn to add, subtract, multiply,
and divide radical expressions, including
binomial radical expressions.
You will solve radical equations, and graph
translations of radical functions and their
inverses.
7-1 Roots and Radical Expressions
What you’ll learn …
 To simplify nth roots

1.01 Simplify and perform operations with
rational exponents and logarithms (common
and natural) to solve problems.
Since 52 = 25, 5 is a square root of 25.
Since 53 = 125, 5 is a cube root of 125.
Since 54 = 625, 5 is a fourth root of 625.
Since 55 = 3125, 5 is a fifth root of 3125.
This pattern leads to the definition of nth root.
Definition nth Root
For any real numbers a and b, any positive
integer n, if an = b, then a is an nth root of
b.




24 = 16 2 is 4th root of 16.
(-2)4 = 16 -2 is 4th root of 16.
x4 = -16
No 4th root of -16.
(√10)4 = 100 4th root of 100 is√10.
Type of
Number
Number of Real
nth Roots when n
is Even
Number of Real
nth Roots when n
is Odd
Positive
0
Negative
2
1
None
1
1
1
Example 1a Finding All Real Roots


Find all square roots of .0001, -1 and
36
.
121
Find all cube roots of 0.008, -1000, and
.
1
27
Example 1b Finding All Real Roots

Find all fourth roots of -.0001, 1 and

Find all fifth roots of 0, -1, and 32 .
16 .
81
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A radical sign is used to indicate a
root.
The number under the radical sign is
the radicand.
The index gives the degree of the
root.
radical sign

When a number has two real roots, the
positive root is called the principal root
and the radical sign indicates the
principal root. The principal fourth root of
16 is written
4
√16

The principal fourth root of 16 is 2 because
4
4
√16 equals √24 . The other fourth root
4
of 16 is written as - √16 which equals -2.
Example 2 Finding Roots
Find each real number root.
3
 √-27
4
 √81
 √49

Notice that when x=5, √x2 = √52 = √25 = 5 =x.
And when x=-5, √x2 = √(-5)2 = √25 = 5 ≠ x.
Property
nth Root of an, a < 0
For any negative real number a,
n n
√a
= a when n is even.
Example 3a Simplifying Radical Expressions
Simplify each radical expression.
 √4x6
3
3
6
 √a b
4
 √x4y8
Example 3b Simplifying Radical Expressions
Simplify each radical expression.
 √4x2y4
3
6
 √-27c
4
 √x8y12
Example 4 Real World Connection
A citrus grower wants to ship a select
grade of oranges that weigh from 8 to 9
ounces in gift cartons. Each carton will hold
three dozen oranges, in 3 layers of 3
oranges by 4 oranges.
The weight of an orange is related
to its
3
d
diameter by the formula w =
, where d
4
is the diameter in inches and w is the weight
in ounces. Cartons can be ordered in whole
inch dimensions. What size cartons should
the grower order?
Find the diameter if w = 3 oz 5.5 oz 6.25 oz.
7-2 Multiplying and Dividing
Radical Expressions
What you’ll learn …
 To multiply radical expressions
 To divide radical expressions

1.01 Simplify and perform operations with
rational exponents and logarithms (common
and natural) to solve problems.
To multiply radicals consider the following:
√16 * √9 = 4 *3 =12 and √16 *9 = √144 = 12
Property
n
Multiplying Radical Expressions
n
If √a and √b are real numbers,
n
n
n
then √a * √b = √ab.
Example 1a Multiplying Radicals
Multiply. Simplify if possible.
 √3 *
√12
3
 √3 *
√-9
4
4
*
 √4
3
√ -4
Example 2 Simplifying Radical Expressions
Simplify each expressions. Assume that all
variables are positive.
•
√50x4
3
•
√18x4
•
3√7x3 * 2√21x3y2
Example 3 Multiplying Radical Expressions
Multiply and simplify.
•
3√7x3
3
*
2√21x3y2
3
2
3
*
• √54x y
√5x3y4
To divide radicals consider the following:
√36 = 6 and 36 = (6)2 = √36
√25
5
25
(5)2
√25
Property
n
Dividing Radical Expressions
n
If √a and √b are real numbers,
n
then
√a
a
= n
n
√b
b
Example 4 Dividing Radicals
Multiply. Simplify if possible.
√243
√12x4
√27
√3x
√1024x15
√4x

To rationalize a denominator of an
expression, rewrite it so there are no
radicals in any denominator and no
denominators in any radical.
Example 5 Rationalizing the Denominator
Rationalize the denominator of each expression.
7
5
√2x3
√10xy
√4
√6x
7-3 Binomial Radical Expressions
What you’ll learn …
 To add and subtract radical expressions
 To multiply and divide binomial radical
expressions

1.01 Simplify and perform operations with
rational exponents and logarithms (common
and natural) to solve problems.
Like radicals are radical expressions
that have the same index and the same
radicand. To add or subtract like
radicals, use the Distributive Property.
Example 1 Adding and Subtracting
Radical Expressions
3
3
5√x -3√x
4 √ xy + 5 √ xy
4√2 -5√3
7 √ 5 - 2 √5
2√7 +3√7
4
3
Example 2 Simplifying Before
Adding or Subtracting
6 √ 18 + 4 √ 8 - 3√ 72
√ 50 + 3 √ 32 - 5 √ 18
Example 4 Multiplying Binomial Radical Expressions
(3 + 2√ 5 ) ( 2 + 4 √ 5 )
(√ 2 - √ 5 )
2
Example 5 Multiplying Conjugates
(2 + √ 3 ) ( 2 - √ 3 )
(√ 2 - √ 5 ) (√ 2 + √ 5 )
Example 6 Rationalizing a Binomial Radical
Denominator
3 + √5
1 - √5
6 + √15
4 - √15
7-4 Rational Exponents
What you’ll learn …
 To simplify expressions with rational
exponents

1.01 Simplify and perform operations with
rational exponents and logarithms (common
and natural) to solve problems.

Another way to write a
radical expression is
to use a rational
exponent.
√25 = 25½
3

Like the radical form,
the exponent form
always indicates the
principal root.
√27 = 27⅓
4
√16 = 161/4
Example 1 Simplifying Expressions with
Rational Exponents

1251/3
P/R = power/root

5½
r
√x p
( √x )p
r
 2½

*
2½
2½ * 8½
A rational exponent may have a numerator
other than 1. The property (am)n = amn
shows how to rewrite an expression with
an exponent that is an improper fraction.
Example
253/2 = 25(3*1/2) = (253)½ = √253
Example 2 Converting to and
from Radical Form

x3/5

y -2.5

y -3/8

√a3

( √b )2

5
3
√x2
Properties of Rational Exponents
Let m and n represent rational numbers. Assume that no denominator = 0.
Property
am * an = a m+n
Example
8⅓ * 8⅔ = 8 ⅓+⅔ = 81 =8
(am)n = amn
(5½)4 = 5½*4 = 52 = 25
(ab)m = ambm
(4 *5)½ = 4½ * 5½ =2 * 5½
Properties of Rational Exponents
Let m and n represent rational numbers. Assume that no denominator = 0.
Property
Example
a-m = 1
am
9 -½
am
a m-n
=
an
1
1
=
=
½
9
3
π3/2
π 3/2-1/2 = π1 = π
=
½
π
⅓
a
b
m
=
am
bm
5
27
=
5⅓
27 ⅓
5⅓
3
Example 4 Simplifying Numbers
with Rational Exponents
(-32)3/5
4 -3.5
Example 5 Writing Expressions
in Simplest Form
(16y-8) -3/4
(8x15)-1/3
7-5 Solving Radical Equations
What you’ll learn …
 To solve radical equations

2.07 Use equations with radical expressions to
model and solve problems; justify results. a)
Solve using tables, graphs, and algebraic
properties.
A radical
equation is an
equation that
has a variable
in a radicand
or has a
variable with
a rational
exponent.
Radical Equation
Not a Radical Equation
Steps for Solving a Radical Equation
1.
Get radical by itself.
2.
Raise both sides to index power.
3.
Solve for x.
4.
Check.
Example 1 Solving Radical
Equations with Index 2
Solve
2 + √3x-2 = 6
√5x+1 – 6 = 0
Example 2 Solving Radical
Equations with Rational Exponents
Solve
2 (x – 2)2/3 = 50
3(x+1)3/5 = 24
Real World Connection
A company manufactures solar cells that
produce 0.02 watts of power per square
centimeter of surface area. A circular
solar cell needs to produce at least 10
watts. What is the minimum radius?
Example 4 Checking for
Extraneous Solutions
Solve
√x – 3 + 5 = x
√3x + 2 - √2x + 7 = 0
Example 5 Solving Equations
with Two Rational Exponents
Solve
Solve
(2x +1)0.5 – (3x+4)0.25 = 0
(x +1)2/3 – (9x+1)1/3 = 0
7-8 Graphing Radical Functions
What you’ll learn …
 Graph radical functions

2.07 Use equations with radical expressions to
model and solve problems; justify results. a)
Solve using tables, graphs, and algebraic
properties.
A radical equation defines a radical function.
The graph of the radical function y= √x + k is
a translation of the graph of y= √x. If k is
positive, the graph is translated k units up. If
k is negative, the graph is translated k units
down.
Example 1 Translating Square
Root Functions Vertically
y = √x
y = √x + 3
Example 2 Translating Square
Root Functions Horizontally
y = √x
y = √x + 3
Example 3 Graphing Square
Root Functions
y = -√x
Example 4 Graphing Square
Root Functions
y = -2√x+1 - 3
Real World Connection
The function h(x) = 0.4 3√ x models the height h
in meters of a female giraffe that has a mass
of x kilograms. Graph the model with a
graphing calculator. Use the graph to estimate
the mass of the young giraffe in the photo.
2.5 m
Example 6
Graphing Cube Root Functions
3
y = 2√x+3 - 1
Example 7 Transforming Radical Equations
Rewrite y = √4x-12 to make it easy
to graph using a translation. Describe
the graph.
Rewrite y = √8x-24 +3 to make it
easy to graph using a translation.
Describe the graph.
3
In this chapter, you should have …



Extended your knowledge of roots to
include cube roots, fourth roots, fifth roots,
and so on.
Learned to add, subtract, multiply, and
divide radical expressions, including
binomial radical expressions.
Solved radical equations, and graphed
translations of radical functions and their
inverses.