Monday, April 28, 2014
Algebra 2 GT
Objective: We will explore and define ways
to simplify radical expressions.
Warm Up: Rewrite each as an exponential
expression with the smallest possible base.
1.
729

3
1. 729
2.2. 6464
2
3.
4.
6
 3125
3.  31255
 5
6
 343
4.  343
5
 3
Monday, April 28, 2014
Algebra 2 GT
Complete the “7.1 Review”
worksheet, #1 – 12 all.
Check answers to the Exp/Log
Review Packet
Unit Test on Wed 4/30
Roots of Real
Numbers and Radical
Expressions
Sections 7.1 and 7.4 – Facts and
Examples
radical sign
1. index
n
a
radicand
2. The positive root of a number is known
as the PRINCIPAL root. So, the principal
fourth root of 16 is 2 (because 24 = 16).
3. What do you notice?
4. Even roots will only yield positive
answers, and we ensure this by using
the absolute value bars.
5. Odd roots can be positive or negative.
Why is this?
Because
2 8 
3
3
8 2
Even though 2  4, 4 2 because we
will only work with the principal roots.
2
2
6. What operation is associated with taking a
root? In other words, what is the underlying
math involved in this simplification?

3
x y x y
6
21
2
7
Therefore, another way to express roots is
using RATIONAL (fraction) EXPONENTS.
n
a a
1
n
n
and
a 
m
 a  a
n
m
m
n
7. Do you recall these old exponent rules?
a a  a
m
n
mn
m
a
mn

a
n
a
 a   a
m
n

mn
ab  a b
m
m
a 
a
   m
b 
b
 m
1
a  m
a

m
m
m
8. Here’s a hint for simplifying with rational
exponents: Always rewrite numbers in
exponential form, using the smallest base
possible.
3
3
27 =
2
4
625 = 25 = 5
64 =
2
8
=
3
4
6
=2
Definition of
th
n
Root
For any real numbers a and b
and any positive integers n,
n
if a = b,
then a is the nth root of b.
** For a square root the value of n is 2.
Notation
index
4
Radical
sign
81
radicand
Note: An index of 2 is understood but not
written in a square root sign.
4
Simplify
81
To simplify means to find x
in the equation:
4
x = 81
Solution:
4
81 = 3
Principal Root The nonnegative root of a number;
only use the positive value when the
square root symbol is given; however,
use both square roots when you choose
to take the square root (as part of
solving) or when the plus/minus symbol
is given (as shown on the next slide).
64
Principal square root
 64
Opposite of principal
square root
 64
Both square roots
Examples


1.  169 x
4
2. -
13 x 
2 2


8
x
3


4
 13x
  8 x  3 
2 2
   8 x  3
2
2
Examples
3.
4.
3

125 x
3
6
3
m n 
3
3
5 x 
3
2
3
 5x
2
 mn  mn
3
Taking
th
n
roots of variable
expressions:
Using absolute value signs
If the index (n) of the radical is
even, and the power under the
radical sign is even, yet the
resulting power is odd, then we
must use an absolute value sign.
Examples
Odd
Even
1.
4
Even
Odd
Even
2.
 an   an
4
6
 xy 
6
2
Even
xy
2
Odd
Even
3.
2
x
6
 x
Even
3
Odd
Even
4.
6
3  y 
2 18
 3 - y
3
2
Even
 3 - y

3
2