P3.
Radicals and Rational
Exponents
Ch. P3: Radicals and Rational
Exponents
SIMPLIFY
𝟏 𝟑 −𝟒
( 𝒙 )
𝟐
𝟑𝒙−𝟑 (𝟒𝒚𝟐 )𝟑
(𝟐𝒙𝟐 )−𝟏
EVALUATE using order of operations (Ex #1)
𝟔𝟒
𝟗 + 𝟏𝟔
𝟐𝟓 × 𝟒
− 𝟐𝟓
𝟗 + 𝟏𝟔
𝟐𝟓 × 𝟒
𝟑𝟔
𝟒
𝟑𝟔
𝟒
Definitions
• Radical sign
• Radicand
• Radical Expression
• Principal Square root
• Simplify a square root
• Conjugates
RULES for Square
Roots
The Product
Rule
The Quotient
Rule
Adding and
Subtracting
• If a and b represent nonnegative real
number, then
• If a and b represent nonnegative real
numbers and b does not equal 0, then
You may only add “like” radicals
Square Roots of Perfect Squares
The principal square root of a2 is the
absolute value of a.
2
a a
In other words, the principal square root of a2 is the positive root!
Ex:
(4)2
(−4)2
Simplify
with
Product &
Quotient
rule:
To simplify means ______________
Ex#2
a.
500
b.
6𝑥 × 3𝑥
b.
48𝑥 3
6𝑥
Ex #3
a.
100
9
YOU TRY:
75
121
36
5𝑥 × 10𝑥
32𝑥 5
2𝑥
Adding/
Subtracting
Like
Radicals
Ex#4
a. 7 2 + 5 2
b.
5𝑥 − 7 5𝑥
Ex #5
a. 7 3 + 12
b. 4 50𝑥 − 6 32𝑥
YOU TRY:
a. 8 13 + 9 13
b.
17𝑥 − 20 17𝑥
a. 5 27 + 12
b. 6 18𝑥 − 4 8𝑥
APPLY: •
Simplify
•
12
28 + 3 7
18 ( 16 + 18 12
Let’s go back to yester-year….
Evaluate the following:
4
5
THE TRICK?
Evaluate the following:
4
5
RATIONALIZE THE
DENOMINATOR
Rationalize Ex #6
5
The
•
3
Denominator
•
6
12
Ex #7
Rationalize •
with
Conjugates
7
5+ 3
•
8
4+ 5
The End (Day 1)
HW: pg. 42 #1-53 odd
Radicals and Rational
Exponents (DAY 2)
OTHER POWERS
QUESTION OF THE DAY
• WHAT RULES DO YOU ALREADY KNOW ABOUT
EXPONENETS? (WARM UP RELATED TO THIS)
• THEN Discovery activity (next slide)
• HOW ARE RADICALS AND RATIONAL
EXPONENTS RELATED TO EACH OTHER?
Use your calculator to complete the table below:
Expression
Numerical Value
Expression
41/2
2
641/3
Numerical Value
41 = 4
3
82/3
3
25−1/2
2
(23 )1/2
641 =
82 =
25−1 =
2
1
25
23 =
1. What do you notice about your answers to the problems in the same row?
2. Is there some pattern that relates the two expressions in each row to one another?
Describe the pattern.
3. Given the expression (53 )1/4 , what expression using a root symbol would yield the
same numerical value?
3
4. Given the expression 54, what expression utilizing a fractional exponent would yield
the same numerical value?
• Rational exponents
Definitions
• Index
• 𝑎
𝑚/𝑛
=
• Also, if 𝑎
𝑛
𝑎𝑚
−𝑚/𝑛
=
=
𝑛
𝑎
𝑚
1
𝑎𝑚/𝑛
Ex #8
3
Simplify
•
Higher Roots
24
4
•
4
8 4
•
4
81
16
•
3
250
YOU TRY:
3
•
40
•
5
5
8 8
•
3
125
27
•
4
405
Ex #9
3
Combining • 5 16 − 11 3 2
Cube Roots
You try:
3
3
• 3 81 − 4 3
(Refer back to warm up)
Using
Ex #10
definition of
1
1
3
2
•
64
125
m/n
a
1
4
−16
Ex #11
• 27
2
3
9
3
2
81
−
3
4
Ex #12
Simplifying
1/2)(7x3/4)
•
(5x
Expressions
32𝑥 5/3
with
•
16𝑥 3/4
Rational
Exponents