Download Complex Numbers (a + bi)

7.1, 7.2 & 7.3 Roots and Radicals
and
Rational Exponents
• Square Roots, Cube Roots & Nth Roots
• Converting Roots/Radicals to Rational Exponents
• Properties of Exponents Apply to Rational Exponents Too!
•Simplifying Radical Expressions
Multiplying
Dividing
Try graphing : y =
3
 x-1
And
y=

x
Square Roots & Cube Roots
A number b is a square root
of a number a if b2 = a
A number b is a cube root
of a number a if b3 = a
25 = 5 since 52 = 25
3
Notice that 25 breaks down into 5 • 5
So, 25 =  5 • 5
Notice that 8 breaks down into
3
2 • 2 • 2 So, 8 =  2 • 2 • 2
See a ‘group of 2’ -> bring it outside the
radical (square root sign).
See a ‘group of 3’ –> bring it outside
the radical (the cube root sign)
Example: 200 = 2 • 100
= 2 • 10 • 10
= 10 2
Note: -25 is not a real number since no
number multiplied by itself will be negative
8 = 2 since 23 = 8
3
3
Example: 200 = 2 • 100
3
= 2 • 10 • 10
3
= 2 • 5 • 2 • 5 • 2
3
= 2 • 2 • 2 • 5 • 5
3
= 2 25
3
Note: -8
IS a real number (-2) since
-2 • -2 • -2 = -8
Nth Root ‘Sign’ Examples
Even radicals of positive numbers
Have 2 roots. The principal root
Is positive.

16 = 4 or -4

-16 not a real number

-16 not a real number

-32
= -2
Odd radicals of negative numbers
Have 1 negative root.

32
= 2
Odd radicals of positive numbers
Have 1 positive root.
4
5
5
Even radicals of negative numbers
Are not real numbers.
Exponent Rules
(x )  x
m n
mn
x 1
0
(XY)m = xmym
x x  x
m
n
m n
m
x
m n

x
n
x
m
X
Y
Xm
= m
Y
x
m
m
1
 m
x
x x
1/ m
Examples to Work through
3
27 
4
81 
12 
3
8x y 
4
3
Product Rule and Quotient Rule
Example
8
8
5/ 4
1/ 4
8
3/ 4

Some Rules for Simplifying
Radical Expressions
n
a  b  ab
n
a a
n
a a
n
m
n
1/ n
m/n
Example Set 1
3
5 7 
5
2 x 5 16 y


y
x
3
300 
Example Set 2
3
16 
54 
4
512 x 
4
Example Set 3
3
5t  3 125t 
6
5
4t 5 8t
 6 
r
r
5
4 5 8


 9  27
7.4 & 7.5: Operations on Radical
Expressions
•Addition and Subtraction (Combining LIKE Terms)
•Multiplication and Division
• Rationalizing the Denominator
Radical Operations with
Numbers
3 2 4 2 
2 16  5 54  10 2 
3
3
3
Radical Operations with
Variables
3
3
8x 2 x


27
3
4
xy  x y 
5
4
5
2 3z  3 12 z  3 48 z 
Multiplying Radicals (FOIL
works with Radicals Too!)
( 2 x  3 y )( 2 x  3 y ) 
( x  9)( x  8) 
Rationalizing the Denominator
• Remove all radicals from the denominator
1

2
xy
y
3

Rationalizing Continued…
• Multiply by the conjugate
1

32
3

32
7.6 Solving Radical Equations
#1
X2 = 64
#2
(3x  6)  25
2
#3
x  1000
#4
(4  x)  1000
3
3
Radical Equations Continued…
Example 2:
Example 1:
x + 26 – 11x = 4
26 – 11x = 4 - x
(26 – 11x)2 = (4 – x)2
26 – 11x = (4-x) (4-x)
26 - 11x = 16 –4x –4x +x2
26 –11x = 16 –8x + x2
-26 +11x -26 +11x
0 = x2 + 3x -10
0 = (x - 2) (x + 5)
x – 2 = 0 or x + 5 = 0
x=2
x = -5
3x + 1 – x + 4
=1
3x + 1 = x + 4 + 1
(3x + 1)2 = (x + 4 + 1)2
3x + 1 = (x + 4 + 1) (x + 4 + 1)
3x + 1 = x + 4 + x + 4 + x + 4 + 1
3x + 1 = x + 4 + 2x + 4 + 1
3x + 1 = x + 5 + 2x + 4
-x -5 -x -5
2x - 4 = 2x + 4
(2x - 4)2 = (2x + 4)2
4x+16
4x2 –16x +16 = 4(x+4)
4x2 –20x = 0
4x(x –5) = 0, so…4x = 0 or x – 5 = 0
x = 0 or x = 5
7.7 Complex Numbers
REAL NUMBERS
Rational
Numbers
(1/2 –7/11, 7/9, .33
Integers
(-2, -1, 0, 1, 2, 3...)
Whole Numbers
(0,1,2,3,4...)
Natural Numbers
(1,2,3,4...)
Irrational
Numbers
, 8, -13
Imaginary Numbers
Complex Numbers
(a + bi)
Real Numbers
a + bi with b = 0
Rational
Numbers
Integers
Whole Numbers
Natural Numbers
Irrational
Numbers
Imaginary Numbers
a + bi with b  0
i = -1
where
i2= -1
Simplifying Complex Numbers
A complex number is simplified if it is in standard form:
a + bi
Addition & Subtraction)
Ex1: (5 – 11i) + (7 + 4i) = 12 – 7i
Ex2: (-5 + 7i) – (-11 – 6i) = -5 + 7i +11 + 6i = 6 + 13i
Multiplication)
Ex3: 4i(3 – 5i) = 12i –20i2 = 12i –20(-1) = 12i +20 = 20 + 12i
Ex4: (7 – 3i) (-2 – 5i) [Use FOIL]
-14 –35i +6i +15i2
-14 –29i +15(-1)
-14 –29i –15
-29 –29i
Complex Conjugates
The complex conjugate of (a + bi) is (a – bi)
The complex conjugate of (a – bi) is (a + bi)
(a + bi) (a – bi) = a2 + b2
Division
7 + 4i
2 – 5i
2 + 5i =
2 + 5i
14 + 35i + 8i + 20i2 = 14 + 43i +20(-1)
4 + 10i –10i – 25i2
4 –25(-1)
14 + 43i –20 = -6 + 43i = -6
43 i
+
4 + 25
29
29
29
Square Root of a Negative
Number
25 4 = 100 = 10
-25 -4 = (-1)(25) (-1)(4)
Optional Step
= (i2)(25) (i2)(4)
= i 25 i 4
= (5i) (2i) = 10i2 = 10(-1) = -10
Practice – Square Root of Negatives
1  i
4 
 16 
 12 
Practice – Simplify Imaginary
Numbers
i0 =
i1 =
1
i
Another way to calculate in
i2 =
-1
i3 =
-i
Divide n by 4. If the remainder is r
then in = ir
i4 =
1
Example:
i11 = __________
i5 =
i
11/4 = 2 remainder 3
i6 =
-1
So, i11 = i3 = -i
Practice – Simplify More Imaginary
Numbers
i 
15

i
26
i
100

i
203

Practice – Addition/Subtraction
(3  9i )  (7  i ) 
10 +8i
(3  9i )  (7  i ) 
-4 +10i
Practice – Complex Conjugates
• Find complex conjugate.
5  2i 
3  4i 
3i
=>
-4i
=>
Practice Division w/Complex
Conjugates
7  4i

4i
4__
2i
=
Things to Know for Test
1.
Square Root, Cube Root, Nth Root - Simplify
2.
Rational Exponents – Convert back and forth to/from radical form
3.
Add, Subtract, Multiply & Divide radicals & rational exponents
4.
Rationalize denominator
5.
Solve radical equations
6.
Imaginary Numbers – Add, subtract, multiply, divide
7.
Imaginary Numbers – find the value of in