Algebra 2B: Chapter 6 Notes
1
Chapter 6: Radical Functions and Rational Exponents
Concept Byte (Review): Properties of Exponents
Recall from Algebra 1, the Properties (Rules) of Exponents.
Property of Exponents: Product of Powers
xm  xn = xm + n
a)
(5x7)(x6)
b)
3r4(-12r3)
c)
6cd5(5c2d2)
c)
(k4)5
c)
(9h2x4)2
Property of Exponents: Power of a Power
(xm)n = xmn
a)
(x5)9
b) [(23)3]2
Property of Exponents: Power of a Product
(xy)m = xmym
a)
(2x2y)4
b)
(8g3h2)3(5g2h2)4
Dividing Monomials:
Property of Exponents: Quotient Rule
xm
 x mn
n
x
a)
x9
x4
b)
x3 y8
xy 2
c)
x5c9
x 2c 4
Algebra 2B: Chapter 6 Notes
2
Zero Property of Exponents
x0  1



a)
x

y 
0
b)
  2x 2

3
 7y



b)
 x 4 y 3 
 5 8 
x y 
0
c)
t 3s0
t
c)
x 3 y 0 x 7
Property of Negative Exponents
xm 
1
xm
5 2
a)
Property of Exponents: Power of a Quotient
m
 x
xm
   m
y
 y

a) 

x

y 
5
 5b 2 n 4
b)  2 3
 n z



1
 3
c)  
2
3
Algebra 2B: Chapter 6 Notes
3
Example: Simplify and rewrite each expression using only positive exponents.
1. (5a 3 )(3a 4 )
3.
5.
4ab 6 c 3
a 5bc 3
2.
(4 x 3 y 5 ) 2
4.
 3a 2b 
 3 
 b 
1
Algebra 2B: Chapter 6 Notes
4
6.1 Roots and Radical Expressions
32  9
We say “3” is the ________ root of 9 and write this:
43  64
We say “4” is the _______ root of 64 and write this:
24  16
We say “2” is the _______ root of 16 and write this:
When we talk about roots:
odd roots  3 a , 5 a , etc.
are positive is a is positive and negative if a is negative.
Example:
3
1000
3
8
even roots  a , 4 a , 6 a , etc.
are only possible “REAL NUMBERS” if a is positive.
We will also only consider the “positive” root/ also called the “principle root.”
Example:
100
4
625
49
Algebra 2B: Chapter 6 Notes
5
Example 1:
Find the real cube roots of each number.
Example 2:
Find the real fourth roots of each number.
a)
a) 16
0.008
b) -1000
b)
-0.0001
c)
c)
Example 3:
Find the real fifth roots of each number.
Example 4:
Find the real square roots of each number.
a) 0
a) .01
b) -1
b) -25
c) 32
c) 36
16
81
121
Example 5: Find each real number root.
a)
3
125
b)
4
81
c)
 7 
2
d)
72
“Radical Expressions”
Or, what happens when variables get involved.
** Note ** in your book, the authors take a lot of time trying to help you understand that when variables are
involved, you need to be especially careful about the ideas of positive/negative. The authors really want you to
use absolute value signs to “force” a variable to be positive. For the purposes of our class, we will be writing our
answers without these absolute value signs.
Algebra 2B: Chapter 6 Notes
6
Recall from “Rules of Exponents”
x 
2 3
By this same logic: Simplify each radical expression.

3
x6 
(ab) 4 
4
a 4b 4 
 2xy 
3 2
4x 2 y 6 

Example 6: Simplify each radical expression
a)
16x8
b)
3
a6b9
c)
4
x8 y12
d)
81x4
c)
3
125a12b15
e)
4
x12 y16 z 8
Example 7:
You can use the expression D = 1.2 h to approximate the visibility range D,in miles, from a height of h feet
above ground. How far above ground is an observer whose visibility range is 84 miles?
Algebra 2B: Chapter 6 Notes
7
6.2 Multiplying and Dividing Radical Expressions
We’ve talked a little about simplifying numerical expressions as we have solved quadratic equations using the
quadratic formula and by square roots. We will use these same ideas to multiply radical expressions.
If a and b are positive,
Recall:
then a  b  a  b.
Write in simplest radical form:
200
75
48
Example 1: Can you simplify the product of the rational expression?
a)
b)
If the radicand has a perfect root among its factors, you can used the product rule to simplify.
This is called simplest radical form and is NOT A CALCULATOR ESTIMATE.
Example 2: Write in simplest radical form.
a)
3
250
b)
5
160
c)
4
162
d)
3
56
Algebra 2B: Chapter 6 Notes
8
We can still use our “division” idea to deal with variable exponents, but now let’s consider expressions that don’t
divide evenly.
Yesterday:
Example 3:
a)
4
a 20
Today:
4
a14
Think: 4 14
x7 y9 z
c)
Write in simplest radical form.
3
a3b8
b)
Simplifying a Product:
Step 1: Use the product rule to combine like radicals
Step 2: Simplify using perfect Nth factors.
Example 4: What is the simplest form of the expression?
a)
b)
c)
d)
4
x 2 y 3  4 x5 y
5
xy 9
Algebra 2B: Chapter 6 Notes
9 2x  3 7 y
e)
g)
Quotients: Dividing Radicals
Example 5: Simplify the following quotients
18 x 5
a)
2 x3
3
b)
162 y 5
3
c)
3y2
50 x 6
2 x4
9
f)
4 2x  3 8x
h)
7 3 y 2  2 6 x3 y
Algebra 2B: Chapter 6 Notes
For a radical expression to be “simplified”
 No perfect square factors under radicals
 No radicals in denominators
 No denominators under radicals
10
A frequent simplification issue:
14
28
To solve this simplification problem we are going to
RATIONALIZE THE DENOMINATOR!
Rationalize the denominator:
 Multiply the fraction by something equivalent to 1.
(The same value to the top and bottom…)
 Goal: Create a perfect square/ perfect nth factor on the denominator.
Example 6: Rationalize the denominator of each expression:
a)
x
6
b)
9x
2
18 x 2 y
c)
2 y3
7 xy 2
3
d)
3
x
8y
e)
f)
4x2
4
3a
4b 2 c
Algebra 2B: Chapter 6 Notes
11
6.3 Binomial Radical Expressions
“Like Radicals” are radicals with the same index and the same radicands.
You can multiply and
divide any radicals
with the same index.
HOWEVER,
You can only add
and subtract LIKE
RADICALS.
Be especially cautious when combining/adding radicals.
3  3  1.73  1.73  3.46
Example:
2  3  1.41  1.73  3.14
2 3  2(1.73)  3.46
5  2.24
3 3 2 3
2 3 5
Example 1: What is the simplified form of each expression?
a) 3 5 x  2 5 x
b)
6x2 7  4x 5
c) 12 3 7 xy  8 5 7 xy
d) 7 3 5  4 5
e) 3x xy  4x xy
f) 17 5 3x2 15 5 3x2
Sometimes you may have like radicals, but you can’t “see” them until you simplify.
Example 2: What is the simplest form of the radical expression?
a)
12  75  3
b)
3
250  3 54  3 16
Algebra 2B: Chapter 6 Notes
12
Sometimes you have to use FOIL to simplify a radical expression.
Example 3: What is the product of each radical expression?

a) 3  2 5
c)
6 
 2  4 5 
b)
3  7 5  7 

d)
3  8 3  8 
12 6  12

Notice that in parts (c) and (d) that you are multiplying CONJUGATES:
a  b and a  b
Any time you multiple radical conjugates, the result is a rational number.
Example:
Here our denominator is: _____________ so we want to multiply by its conjugate ______________.
5
2 3
Algebra 2B: Chapter 6 Notes
13
Example 4: Write the expression with a rationalized denominator.
2 7
3 5
a)
b)
6.4 Rational Exponents
Check the following in your calculator:
49
1
2
125
1
121
3
1
1
81 4
2
16
1
32
2
1
5
4x
3 6
Algebra 2B: Chapter 6 Notes
n
a 
a
m
14
m
n
Dealing with Rational Exponents:
1. Rewrite the expression as a radical.
2. Multiply if necessary using radical rules.
3. Simplify if you can.
Example 1:
a) 64
1
1
b) 11 2 11
2
1
1
c) 3 2 12
2
Example 2: Rewrite in simplest radical form
a) x
3
5
c) w
b) x0.2
7
8
d) y 3.5
Example 3: Rewrite in exponential form.
a)
a5
b)
5
b3
c)
4
x2
d)
 y
5
4
1
2
Algebra 2B: Chapter 6 Notes
Example 4: What is each product or quotient in simplest form.
a)
4
x3
8
x2
b)
c)
d)
3
 3
4
x3
3
x2
7
 7
3
15
Algebra 2B: Chapter 6 Notes
Example 5: What is each number in simplest radical form?
a)
93.5
b)
81 4
c) 32
3
3
5
d) 162.5
Example 6: Simplify each expression.
15
a) (8 x )
b)

1
3
16
Algebra 2B: Chapter 6 Notes
17
6.5 Solving Radical Equations
Any time you have a variable under the radical sign, you may have to use exponents to solve.
Solve: 3  2 x  3  8
Steps
1. Isolate the radical
2. Raise each side of the equation to the nth powers
3. Solve the equation
4. CHECK YOUR ANSWERS!!!!
Example 1:
a)
c)
4x 1  5  0
3 5 ( x  1)3  1  25
b)
d)
5x  1  3  x
3( x  1)
2
3
 12
Algebra 2B: Chapter 6 Notes
18
3x  1  2 x  7  0
e)
f)
2 x  2  10  4
6.8 Graphing Radical Functions
Graph the following radical expression
y
y  x3
x

y
















y

x


















x






































Algebra 2B: Chapter 6 Notes
19
y  x 2
y  2 x  1

y
















y

x


















x




























y  3 x4


















y  3 x  4

y
















y

x



















x




























