Transcript 7. Roots and Radical Expressions

7. Roots and Radical
Expressions
In this chapter, you will learn:
•
•
•
•
What a polynomial is
Add/subtract/multiply/divide polynomials
Simplify radicals, exponents
Solving equations with exponents and
radicals
• Complex numbers
• Conjugates
What is a monomial?
An expression that is a
number, that may or may not
include a variable.
MONOMIALS
 2x
2
10 xy
4x
NOT MONOMIALS
4
x
8
x8
1
5x
Real Roots
• Real roots are the
possible solutions to a
number, raised to a
24  (2) 4  16
power.
x2  x  x
or   x   x
so bot h xand - x are
root sof x t ot hesecond
power
so both 2 and negative2 are
rootsof 16 to thefourthpower
 53  125
so - 5 is theonly possible
real rootof - 125
(why can't it be  5?)
Vocabulary and Properties
Radical sign
index
n
a
radicand
How to find the root (other than a square
root), using a graphing calculator
1. Input the root you are going to
take (for example, if you are
taking the third root of a
number, start with the 3).
2. Press MATH and select option
5 x
3. Enter the value you are taking
the root of.
Ex: 4 81  4 MATH 5 81 ENTER
3
Practice: Find each root
3
5
4
10648
16807
16
Solutions: 22,
7, and
ERR:
NONREAL
ANS
Let’s take a closer
look at this answer
Properties and Notation:
n
aa
When n is an even
number
Why? We want to
make sure that the
root is always positive
when the index is an
even number
Here's another way tolook at it :
IF x  5, x 2  52  25  5  x
BUT if x  5, x 2  (5) 2  25  5  x
so we want tomakesure that xis always a positive
with even indexesso thatit is true backwards
and forwards.We use theabsolute value to makesure it
stays positive.
ex :
4
16x y  (2) x ( y )  2 x y
4
8
4
4
4
2 4
Note: Absolute value symbols ensure that the
root is positive when x is negative. They are not
needed for y because y2 is never negative.
ex :
Notice
that the
index is
an odd
number
here . . .
3
x y  x (y )  x y
3
6
3
3
2 3
2
Absolute value symbols must not be used here.
If x is negative, then the radicand is negative
and the root must also be negative.
2
Let’s try some
Simplify each expression. Use the absolute value symbols when needed.
2
4x y
4
3
 27 x y
6
Solutions
Simplify each expression. Use the absolute value symbols when needed.
2
4x y
4
3
 27 x y
6
Properties of Exponents – let’s review . . .
NEGATIVE EXPONENT
RULE
5
a
2
n
1
 2
5
1
 n
a
PRODUCT OR POWER
RULE
2 2
10
20
2
30
m n
a a  a
m
n
HAVE TO HAVE THE
SAME BASE
QUOTIENT OF POWER
RULE
10
3
6

3
4
3
a
x
a b

x
b
x
HAVE TO HAVE THE
SAME BASE
POWER OF POWER
RULE
4
x )³
(
x
43
x
(a )  a
m n
mn
12
POWER OF PRODUCT
RULE
4
2x )⁵
(
2 x
5
45
 32x
(ab )  a b
m n
n mn
20
POWER OF A QUOTIENT
RULE
3
3I
F

G
J
Hy K y
5
5
5
aI a
F

G
J
Hb K b
n
n
n
POWER OF QUOTIENT 2
RULE
3
 
y
2
3  y   y 
   2    2    
3
9
y






2
x
 
y
a
2
y 
  a 
x 
a
2
Fractional Exponents
(Powers and Roots)
“Power”
x
y
a  a ( a )
y
“Root”
x
y
x
RADICAL TO EXPONENT
RULE
1/ 2
25
1/ 2
a
 25
 a a
1/ n
5
 a
n
RATIONAL EXPONENT
RULE
16
3
4
m
n

 16 
3
4
a  a 
n
m
 2
3
8
 a
n
m