Download Roots and Radical Expressions

Roots and Radical
Expressions
6-1
Objective
To find the n
th
root.
Key Concept
If a n  b, a and b are both real numbers and n is a positive
integer, then a is the nth root of b.
Radical Sign
Index
n
a
Radicand
n Roots of n
th
For any real number a,
n
th
Powers
a
a 
 a
n
If n is odd
If n is even
Principle Root
There are two real b roots of b. There
is positive b, and negative b.
Finding all real roots
What is the real cube
root of 0.008?
0.008  (0.2)
What is the real fourth
root of -0.0001?
3
Since -0.0001 is negative,
there is no real fourth
root.
What is the real cube
1
root of 27 ?
1
1 3
( )
27 3
What is the real fourth
root of 16 ?
81
16
2 4
 ( )
81
3
Finding Roots:
What is the real-number root?
3
8
0.04
(2)3  8
So,
3
8  (2)
 0.2   0.04
2
So,
0.04  0.2
Finding Roots:
What is the real-number root?
4
1
There is no real root because
there is no real number whose
fourth power is -1.
(2) 2
 2 
2
 42
Simplifying Radical Expressions
16x
4
2
8
x x x x x x x x
4
2 2
2  2  x  x  x  x  4x 4
2
Simplify Radical Expressions:
Questions
3
4
6 9
ab
8
x y
12
Answers
2 3
ab
2
x y
3
STEM
The voltage V of an audio systems speakers can be
represented by V  4 P , where P is the power of the
speaker. An engineer wants to design a speaker with
400 watts of power. What will the voltage be?
Definition of i
i  1
and
i  1
2
The imaginary number, i, was invented so we
can solve equations like:
x  4
x   4
2
Remember, it’s Not a Real Number!
It’s an Imaginary Number!
Properties of i
i  1
i 5  i 4  i1  1 i  i
i 2  1
i 6  i 4  i 2  1 1  1
i 3  i 2  i1 1 i  i
i 4  i 2  i 2  1  1  1
i 7  i 4  i 3  1 i  i
i 8  i 4  i 4  11  1
They repeat the first 4!
You can find any power of i
Simplify
441
Divide the exponent by 4
i
1
i
i
Your answer is i with the
remainder as it’s exponent.
110
4 441
440
1
Memorize the first 4 powers of i:
i i
1
i  1
2
i  i
3
i 1
4
T.O.O. Simplify
1)
3)
73
i
1
i
i
12
i
0
i
1
18
4 73
72
1
3
4 12
 12
0
2)
102
i
2
i
1
25
4 102
100
2
15
3
4 15
 12
3
4)
i
3
i
i
Simplify
Not a real number,
1) 5
but now have new
definition
(1)(5)
i  1
1  5
i 5
Put the i in front
of radical!
Don’t write:
5i
2)
25
1 25
1  25
i 5
5i
i after integer
Homework
Pg 364 # 11-29 Odd, 30, 45-48 ALL