Transcript Lecture Notes for Sections R7 (Castillo

Radicals
Basic meaning
a = x means x × x = a
m
a = x means x = a
m
Examples
2 is the number such that 2 2 = 2
3
4
27 = 3 because 3× 3× 3=27
p is the number such that
( p)
4
4
=p
Important meaning
a =a
m
1
2
a =a
( a)
m
1/m
n
=a
n/m
Why are these true?
• Because I want to keep the rules of
exponents.
• Specifically aman=am+n
• Let’s see how that works?
What should 21/2 mean?
• I can make it mean anything I want.
• But I have a rule that I want to keep:
aman=am+n
• If I try to use this rule on 21/2, I get
• 21/221/2=21/2+1/2=21=2
• So 21/2 is the number that when multilpied
by itself gives me 2:
• The square root of 2!
Similarly
Mystery Number: a
1/m
Rule of Exponents: ( a
So ( a
But
)
1/m m
( a)
m
m
=a
1m
m
m
)
=a =a
1
= a (Slide 2)
So let's pick a
1/m
n
=ma
=a
mn
Why is this important?
• By converting radicals to exponents, you
only have to learn one set of rules:
• Instead of learning rules for radicals and
rules for exponents, just learn rules for
exponents.
Example
( a) ( a)
3
4
Example
( ) ( a)
3
a
= (a
4
)
1/2 3
1/4
a
( )
Example
( ) ( a)
3
a
= (a
4
)
= (a ) (a )
1/2 3
3/2
1/4
a
( )
1/4
Example
( ) ( a)
3
a
= (a
4
)
= (a ) (a )
1/2
3/2
=a
3 1
+
2 4
3
1/4
a
( )
1/4
Example
( ) ( a)
3
a
= (a
4
)
= (a ) (a )
1/2 3
3/2
=a
=a
1/4
a
( )
1/4
3 1
+
2 4
6 1
+
4 4
=a
7
4
Example
( ) ( a)
3
a
4
= (a
)
= (a ) (a )
1/2 3
3/2
=a
=a
=
1/4
a
( )
1/4
3 1
+
2 4
6 1
+
4 4
=a
( a)
4
7
7
4
Convert to an expression using rational exponents:
5 5
3
2/3
a) 5
5/6
b) 5
3/4
c) 5
3/2
d) 5
e) None of the above
Convert to an expression using rational exponents:
5´ 5
3
=5 5
1/2 1/3
2/3
1 1
+
2 3
a) 5
=
5
5/6
b) 5
3 2
+
3/4
6 6
=
5
c) 5
3/2
5/6
d) 5
=5
e) None of the above
Warning!
• We tried hard to keep everything
compatible with the rules of exponents, but
there is one problem we can’t fix: Even
exponents lose information.
Example
Example:
(-2)
= 4
=2
2
Example
If I try to use
rules of exponents
Example:
(-2)
= 4
=2
2
(-2)
2
= éë( -2 ) ùû
2 1/2
= (-2)
= (-2)
æ1ö
2ç ÷
è2ø
1
= -2
Example
If I try to use
rules of exponents
Example:
(-2)
(-2)
= éë( -2 ) ùû
2 1/2
2
= 4
=2
2
= (-2)
Don’t match!
= (-2)
æ1ö
2ç ÷
è2ø
1
= -2
New (unavoidable) rule
m
x = x , when m is even
m
Or (Same thing):
(x )
1/m
m
= x , when m is even
Simplifying square roots of
numbers
A trick to make your life easier
Prime numbers
• A prime number is a natural number > 1
that is only divisible* by 1 and itself
• *Divisible means after you do the division you get a
natural number.
• Prime numbers: 2,3,5,7,11,13,17,19
Prime factorization
• A prime number is a natural number > 1 that is
only divisible* by 1 and itself
• Factoring means turning something into a
multiplication
• A prime factorization is turning a number into a
multiplication of primes.
• Every number has exactly one prime
factorization
• Prime factorizations are completely factored
Prime factorization examples
• 6=2*3=2131
• 128=2*2*2*2*2*2*2=27
• 1500=2*2*3*5*5*5=223153
How to factor into primes
• Learn your list of primes: 2,3,5,7,11,13,17,19
(are the important ones).
• Take your number (1500), divide it by the first
prime on your list as many times as you can:
• 1500/2=750
• 750/2=375
• 375/2=187.5 BAD
• 1500=2*2*375
• Repeat for the next largest prime, until you get 1
Prime factoring 1500
•
•
•
•
•
Next prime is 3
1500=2*2*375
375/3=125
125/3=41+2/3 BAD
1500=2*2*3*125
Prime factoring 1500
•
•
•
•
•
•
Next prime is 5
1500=2*2*3*125
125/5=25
25/5=5
5/5=1 DONE
1500=2*2*3*5*5*5
Note
• Note: With practice, you will find shortcuts.
You don’t always have to start with
smallest. Looking for numbers you
recognize is faster
• Ex: 1500=15*100=3*5*10*10=3*5*2*5*2*5
• 1500=2*2*3*5*5*5
• No matter how you do it, there is only one
prime factorization to get.
Simplifying Radicals
1500
= 2 * 2 * 3* 5* 5* 5
= 2 * 2 5* 5 3* 5
= 2 * 5 3* 5
= 10 15
Using Exponential Notation
1500
1500
= 2 * 2 * 3* 5* 5* 5 = 2 23153 = 2 2315152
= 2 * 2 5* 5 3* 5 = 2 2 52 3151
1 1
1 1
= 2 * 5 3* 5
=2 5 35
= 10 15
= 10 15
Simplifying Radicals
3
1500
= 2 * 2 * 3* 5* 5* 5
3
= 5* 5* 5 2 * 2 * 3
3
3
= 5 2*2*3
3
= 5 3 12
Using Exponential Notation
243
243
= 3* 3* 3* 3* 3
= 35
= 35
= 3* 3 3* 3 3
= 3134
= 3 33
= 3* 3* 3
= 32 31
=9 3
= 32 31
=9 3
=9 3
Divide the inside
power method
-m powers inside
+1 power outside
method
243
Repeated
Multiplication
Method
Using Exponential Notation
3
243
3
243
= 3* 3* 3* 3* 3
= 3
= 3 3* 3* 3 3 3* 3
= 3 3233
= 3* 3 3* 3
= 31 3 32
= 33 9
= 33 9
3
Repeated
Multiplication
Method
3
5
Divide the inside
power method
3
243
= 3 35
= 31 3 32
= 33 9
-m powers inside
+1 power outside
method
Simplify and express as a single
radical:
162  50  8
a)
d)
204
2 2
b) 7 2  2 5
c)
e) None of these
12 2
Simplify and express as a single
radical:
Prime Factoring 162
162 / 2 = 81
81 / 3 = 27
27 / 3 = 9
9/3=3
3/ 3 =1
162 = 2 * 3* 3* 3* 3
162 + 50 - 8
= 23 + 25 - 2
3
=3
2
1 4
2
1 2
2 +5 2 -2
1
1
2
= 9 2 +5 2 -2 2
= 12 2
C
Simplify completely:
2
128c d
4
a) 64 | c | d 2
b)
d) 4
e) None of these
cd 8
8cd
2
2
c)
8cd 2
Simplify completely:
= 128c d
2
= 2 cd
7
2
4
4
= 22 c d
1 6
=2 c d
3
=8 c d
2
2
2
4
1
2
2