Transcript 5-1-1 Rational Exponents & Simplify Radicals

WARM-UP
Find the prime factorization for the
following numbers. Write your
answer as a product of primes.
1.
72
2.
120
5.1a Rational Exponents & Simplify Radicals
Objective: To simplify rational exponents
n
n2
n3
n4
n5
n6
n7
n8
2
4
8
16
32
64
128
256
3
9
27
81
243
729
2187
4
16
64
256
1024
5
25
125
625
6
36
216
1269
7
49
343
2401
8
64
512
9
81
729
Radicals
Root index
n radicand
x2 
x4 
x6 
x8 
x10 
x even 
What should you do if the exponent is not even?
x9 
x even x1 
x5 
x15 
Simplify: All variables are positive.
25x2 y 7 
20x5 y10 
Cube roots: Look for perfect cubes in the coefficient.
How can you determine if the variable is a perfect cube?
3 27 x 6 y10 
3 40x 7 y12z20 
4 32x2 y 4z15 
Rational Exponents
n x  the n th root of x
1
n x  xn
23 = 8
54 = 625
General :
38  8
1
3
2
4 625  625
power
a
root
n a
x  xn  x
1
4
5
Now – Try some fun problems!
2
27 3 
5
64 6 
3
 4 2
2
 8  3
  
9
 
 27 
Remember: Root first makes the number smaller.
3
256 4 

Can you simplify rational exponents?
Assign 5.1a: 17-39 odd, 41-58 all
243
2
5

WARM-UP
Hmmm…do you remember??
1. x2x3 =
3.
a-3
2.
4.
(x2)3 =
x
x
5
2
=
5.1a Answers
42. 16
44. 27
46. 27
48. 1/3
50. 1/8
52. 7/4
54. 4/9
56. 15
58. 12/35
5.1b Simplifying Radical Expressions
Objective: To simplify rational expressions using
exponent properties
Recall the exponent properties.
a +b
xaxb = x
(xa)b =
x ab
a-n
1
an
=
xa
b =
x
x
a b
x0 = 1, x  0
Now try these!
2
3
Ex.1 x x
Ex. 2
1
2
x 
Ex.3 4
2
2 3
3
2
2
3
x
Ex. 4
x
Ex.5 x 2 x
Ex.6 42 4
3
2
3
2
Simplify. All variables are positive.
6

4
Ex.10  8 x y z 


1
2
2
9 3
Ex.7 3( x y )
 24 xy 2 
Ex.8 

8
 3 yx 
2
3
2
3
1
3
4
3
 23 
x 
Ex.9   2
4 x 4 3
Can you simplify rational expressions using exponent properties?
Homework: 5.1: 59-67 odd, 68-78 all, 93, 94, 107, 108
Quiz after 5,3   
5.1b Answers
68.
3
1
2
x 5 y10 z 3
11
70.
b 3
4
a 15
1
72.
y
74.
a3b5
6
1
1
1
76.
r32 s 2
78.
b3
a
94.
t  9.52, d  14,400
108. x3 4  x4 3 on the int erval
0  x 1
5.2 WARM-UP
Simplify:

3 16x2 y8

1
2
27x9y12 
1
3
5.2 More Rational Exponents
Objective: To continue multiplying rational exponents
Multiply:
Ex1. 4x2 (3x3  2x  1) 
Ex2.
5
2
2x 3  4x 3


1
 2x 2

 1  

 2
 2

3
3



Ex 3.  x  3  x  4  



 1
 1

2
2



Ex 4.  x  5  x  5  



3  2
3
 2
Ex5.  x 3  4 2  x 3  4 2  



Multiply:
1  2
1 1
2
 1
Ex6.  x 3  y 3  x 3  x 3 y 3  y 3  



Factor completely:
Ex 7. 4(x – 3)2 + 5x(x – 3) =
Factor with Rational Exponents
Determine the smallest exponent and factor this from all terms.
Ex8.
x
5
x3
1
 x2
2
x3
4
x5
8
x7
2
 2x 3
6
1
 x3
2
 x5
4
 x7

Ex9.
2
1
4x 5 y  8xy 2
Try these:
10
2x 3
5
 5x 3
 12 
6
4
3x 3
2
 x3
2
6x 5
1
 13x 5
4
6
6
6

Last one!
Add: Don’t forget the common denominator!
4
x 
x
4
1
x2
x 
x3
x  1
1
2
 2x  1
1
2

Can you multiplying rational exponents?
Assign 5.2: 3-69 (x3), 77, 81, 97-100
5.2 Answers
6.
20x3  15x 2  10x
1 1
40x 2 y 2
24.
25x 
42.
1
5  6x 2
 16y
1
 15

5  7 
4
t

7
4
x



60.


100.
1
k6
12. a  8a
30.
48.
66.
1
2
 12
t - 125
1
4x  1 3 x  3
x7
x
1
4
18. t  6t
1
2
9
36. a + 27
1
1



3
54.  x  2  x 3  3 



98. x
3
2
5.3 Simplified Radical Form
Objective: To write Radicals in simplest radical form
Properties for radicals: a, b > 0
1. n a b  n a  n b
na
a
2. n  n for b  0
b
b
3. a b  c d  ac bd
Simplify each radical means:
No perfect squares left under the
No perfect cubes left under the
No fractions under the radical
No radicals in the denominator
3
No factors in the
radicand can be
written as powers of
the index.
When you simplified radicals to this point the book said
that all variables were positive.
What if they do not tell us all variables are positive?
x2  x :
When you have an even root and an even exponent in the
radicand that becomes an odd exponent when removed,
you must use absolute value.
x6  x3
The first one needs absolute
value symbols to insure the
answer is positive
x7  x3 x
The second does not because if x
was negative, it could not be
under an even root.
Simplify each: Do not assume variables are positive.
3 18x10 y17z 4 
8 4 x 4 y10z21 
Type 1: Similar to section 5.1
Ex1. 2 4 162x7 y12z 41 
Ex2.
3 48x3y 4z22 
Type 2: No radicals in the denominator.
5 
3
4x 
3y
16 
5
7
34

2 15 
6
4
4
x
3

2
3y
Try these:
4x 4
9xy
4
5

1
3
2x2 8xy 3
9x yz
2
5x2 y 7

3
2x8
8xy
10


How do you know what degree to make the exponents
in the denominator?
Can you write Radicals in simplest radical form?
Assign 5.3: 3-21 (x3), 23-33 odd, 48-69 (x3), 71-77 odd, 85-87
all, 89, 105
GROUP ACTIVITY
Learning Target: Find a set containing 3
equivalent forms of the same
number on the face.
You will work with the 1 or 2 people sitting
beside you. Begin with all of the cards face-up
spread out on the desk. Take turns gathering
sets of 3 cards.
5.3 Answers
6.
8 2
12. 18 3
18.
8 33
48.
4
54.
135x 2
3x
5xy 6yz
66.
3z
60.
86.
6
6
3 10xy 2
2y
3 3 847
feet
22
Review 5-1 to 5-3
Questions?
Remember NO CALCULATOR!
5.1:
 Simplify radicals and rational exponents
Write radical expressions with rational exponents
Evaluate rational exponents
Simplify expressions with rational exponents
5.2:
 Multiply and factor using rational exponents
 Add by making common denominators with rational exponents
5.3:
 Simplify radicals if the variables may not be positive
 No fractions under the radical
 No radicals in the denominator
 Be able to do these for any root
Now let’s try some problems! 
Write using rational exponents:
8 3
2 5
x 
x 
Simplify:
3
2
3
9
  
 16 
16 4 
2 3 27 x2 y 9z14 
1
a4
2 1
x3 y 4
1
x6y

Does
200x 4 y10z15 

 a
1
: x2

3 9x
1
y2

3

2 4 2
1
2
y
 x  y  ?

Multiply:
 1
 1

 a 3  5  3a 3  1  






2
5
2x  2x


1
3
 3x

 5  

Factor :
2
x3
1
 5x 3
2
4x 5
6
2
2xx  5 3
3
 4x
1
x  53
4
16x 5

1
 15 x 5
2
 24x 5

9
Simplify :
4

12
12
3
y
4
2
3

6x
2
4x
4
3
x3
x  22
 3x 
2

1
 4x  22 
1
Assign: Review WS
5.4 – Addition and Subtraction of Radicals
Objective: To add and subtract radicals
We all know how to simplify an equation
such as: 2x +3y – 5x = 3y – 3x
The process for addition and subtraction of radicals is
very similar. To do so you must have the same index
and the same radicand.
5 2 3 5 2 2  3 2 3 5
Lets try some! **You may need to simplify first!
Ex.1
3 18  5 12  27
Ex2.
Ex.3
Ex.4
3
4x 3  3 4x
x  3 48x3 y  5  3 6x 6 y  3 6y3
4 1

3
2
Ex.5 4 x 18  2 8
Can you add and subtract radicals?
Homework: 5.4
54.
3
2
5.4 Answers
Simplify.
WARM-UP
(7  3)(7  3)
(5  3)(5  3)
What did you notice about the
above? These are called
CONJUGATES!
5.5 Multiplication and Division of Radicals
Objective: To multiply and divide radicals
Recall the radical properties we learned earlier in the chapter.
a b  c d  ac bd
Ex.1
4 5  2 15
Ex.3 2 3  6  3  6 
Then simplify if possible.
Ex.2 5 2  8  4 3 
Ex.4

x  3 x  3
Therefore factorable!!!
Ex.5

x  1  3
2
Now for division. Don’t forget to rationalize the denominator!!
Multiply the numerator &
denominator by the
conjugate of the
denominator. Then FOIL.
4
Ex.6
1 3
5 2
Ex.7
4 2
Ex.8
Can you multiply and divide radicals?
Homework: 5.5
3
x 2
5.5 Answers
6. 4200
12. 105  14 3 5
18. 50  10 21
36. x - 22
42.
24. 25a  20 ab  4b
30. 3
48.
2 x 2 y
xy
54.
a  2 ab  b
a b
7 7
6
60. 11 x  5x  2
66.  x  7    x  7  x  7   x  14 x  49
2
4x
5.6a Equations with Radicals
Objective: To solve basic radical equations
Recall: 4x – 5 = 23
+5 +5
4x
= 28
4
4
x
=7
 Locate the variable.
 Undo order of operations to
isolate the variable.
Procedure:
How is 4 x  5  23 similar?
Locate and isolate the radical.
+5 +5
4x
= 28
4
4
( x )2 = ( 7 )2 How do you undo the radical?
x = 49
Always check these answers. When you
square, you may get extraneous roots.
Squaring Property: If both sides of an equation are squared, the
solutions to the original equation are also solutions to the new equation.
*You must square the entire side.
*You must check for extraneous(extra) roots.
BASIC:
Ex1 : 2 x  3  4  8
Ex3 :
4  3  x  3
Ex2 : 4  2x  3  12
Medium:
Ex 4 : (x  3)2 ( x  3 )2
* Isolate the radical on one side.
* Square both sides (the entire side- FOIL)
* Solve the quadratic. (How?)
x2 – 6x + 9 = x – 3
x2 – 7x + 12 = 0
(x – 4) (x – 3) = 0
x = 4, 3 Check both answers – one generally does not work.
You try these:
Ex5 : a  2  a  10  0
Ex7 :
3 x  5  3 2x  7
Ex6 :
3x 3  4
Can you solve basic radical
equations?
Assign: 5-6 to # 35
5.6b
Solve:
Warm - up
t7 t5
5.6b More Solving Radical Equations
Objective: To solve radical equations with radicals on both sides and
identiry extaneous roots
What happens when you have two radicals that you
cannot combine?
x  5  x  2
* Two different roots & something else
*Isolate the more complicated radical on
one side and square both sides. (The entire
side.)
* Isolate the radical that is left and square
both sides again.
You try this one:
x 5 3  x 8
Graphing y 
x
0
1
4
y
0
1
2
What is the domain:
What is the range:
x
How would each change affect the graph?
Give the domain and range for each.
Last
Ex1
324:: One
yyyy3: xyx
x2332x 1
Domain:
Range:
Summary:
How can you make the root open left? Upside down?
Can you solve radical equations with radicals on both sides and identity
extaneous roots?
Assign: Rest of 5.6
5.6b Solutions
42.
48.
54.
56.
58.
60.
4
5, 13
12 x  125
10
10,000
The plume would be smaller if there was a
current.
5.6c Solving Equations with Rational
Exponents
Objective: To solve equations with rational exponents and
understand extraneous roots
* To solve an equation with a rational exponent, you must first
solve for the variable or parenthesis with the rational exponent.
* You must undo the exponent, by taking it to a power that will cancel
the exponent to a 1.
1
3 3 ( )3
3
( )
x 4  x
Ex1 :
4 3
x 8 
4
x1 = 64
Ex2 : 4x
2
3
5  4
How do you know when you should use  for your solution?
When solving an equation and you must take an even root, you must use
x =  answer.
You try these:
Ex3 : 3x  1
3
4
 1  26
Ex5 : x  6 x  8  0
Ex7 :
4
x3
 13x
2
3
 36  0
Ex 4 : 4x
2
5
 36
Ex6 : x  3 x  40
Miscellaneous
Completely factor: x2n – 5xn + 6
Now try:
Ex: 2x4n + x2n - 6
Ex: x2n+1 - 5xn+1 + 6x
Cancel:
x 3n
x2
x 3n  5
xn
x
xn 1
2n  5
Can you solve equations with rational exponents and understand extraneous roots?
Assignment: Worksheet and begin test review.
5.6c Worksheet Solutions
1.
2.
3.
4.
5.
6.
7.
27
16
32
-32
64
1
64
1
16
8.
14
9.
7
3
10.
4
3
11.
12.
13.
14.
81
1024
63, -62
341
15.
15
2
81
32
35 & -29
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
25
27 & -64
49 & 25
25
32,768 & -32
9 & .25
x(2x n  3)( x n  2)
2 x 3 ( x  3)( x  3)
x 1 (2n2n  1)( x 2n  5)
x 2 n1
x 2 n1
x 2n
x n6
5.7a Introduction to Complex Numbers
Objective: To define imaginary and complex numbers and perform
simple operations on each
Complex Numbers (C) : a + bi
Imaginary (Im):
Real Numbers (R): the set of rational and
irrational numbers.
negative
Rational (Q) : any number that
can be written as a fraction
Irrational (Ir) : non-repeating,
non-terminating decimals
What are imaginary numbers?
Integers (I or Z): positive
and negative Whole numbers:
no fractions or decimals
Whole (W) : {0, 1, 2, 3, …}
a = real part
b = imaginary part
Square roots of negative numbers.
no
fractions, decimals or negatives.
Natural (N) : counting numbers
no decimals, fractions, negatives, 0
Symbol? i
Value?
1
 1 i
A complex number is in the form of
a + bi
where a = real part and bi = imaginary part.
A pure imaginary number only has the imaginary part, bi.
**Always remove the negative from the radical first!**
Ex1:
4
Ex2:

4
9
Ex3:
Ex4:
2 8
Ex5:
9
 25
Ex6:
 27
5 5
What is the value of i 2 ?
(
 1 )2  ( i ) 2
-1 = i2
When you get an i2 , always replace it with a -1.
i9 =
i10 =
i11 =
i12 =
i = i
i2 = -1
i3 =
i4 =
i5 =
i6 =
i7 =
i8 =
Ex1: i20 =
Ex2: i30 =
Ex3: i57 =
Ex4: i101 =
Ex5: i12 i25 i-3 =
For 2 complex numbers to be equal, the real parts must
be equal and the imaginary parts must be equal.
Ex1: 3x + 2i = 6 + 8yi
Ex2: 4x – 3 + 2i = 9 – 6yi
Ex3: 5 – (4 + y)i = 2x + 3 – 6i
Can you define imaginary and complex numbers and perform simple
operations on each?
Assign: 1-24 all
5.7a Solutions
2.
4.
6.
8.
10.
12.
14.
16.
18.
7i
-9i
4i 3
 5i 3
-i
i
-1
x = 1 y = -4
x = 2/3 y = -.5
20.
22.
24.
x = -.5 y = -5/3
x = 11/4 y = -2
x = 2/5 y = -4
5.7b Operations on Complex Numbers
Objective: To perform operations on complex numbers
Add/subtract: Add real part to real part and imaginary part to imaginary part
Compare to:
Add:
(4 – 3x) + (2x – 8) = -x - 4
(2 + 4i) + (6 – 9i) =
Subtract: (5 – 3i) – (7 – 5i) =
Multiply: Distribute or FOIL – all answers should be in standard complex form: a + bi
Don’t forget i2 = -1
Ex1: 2i(3 – 4i) =
Ex2: -4i(5 + 6i) =
Ex3: (2 – 3i)(4 + 5i) =
Ex4: (4 – 2i)2 =
This is similar to rationalizing the denominator with
radicals.
Division:
Type 1:
4
i
5
2i
-or-
4
Recall:
How did we rationalize the denominator?
2 3
Use the complex conjugate to divide complex numbers. a + bi
Type 2:
4
3  2i
-or-
12  8i
9  4i
Can you perform operations on complex numbers?
Assign: 5.7b: 25-77 odd 87-90
a - bi
5.7 b Solutions
88. i
if n is even
90. x = 1 – i is a solution to the equation.
5.7c and Review
Objective: To factor and simplify
using complex numbers
FOIL:
(x - 3)(x + 3)
-compare to-
(x – 3i)(x + 3i)
This means the following can be factored. How?
1. 4x2 - 25
2. 4x2 + 25
3. x2 + 4
4. 2x2 + 98
Just for fun. (And they make great essay questions.)
*What are imaginary numbers?
*What symbol is used to designate imaginary units?
*What is the value of the imaginary unit?
*Give the definition of a complex number?
*What is the complex conjugate and when should it be used? Give an example.
*What is a pure imaginary number?
Ex1 : i2  i30  i5
Ex2 : i13  i21  i8  i50
Ex4: 4x – 3 + 2yi = x + 2y – 8i
Ex7 :
2i
4  3i
Ex10 : 5i2 3i2i
Ex5:
Ex3 : i5  i12  i22  i30
8 6
Ex8 : 4  5i  8i   5  6i
Ex11 : Solve : x2  4  0
Ex6 :
4 i
3i
Ex9 : 3  2i2
Can you factor and
simplify using complex
numbers?
Assign Worksheet:
Mini-Quiz Tomorrow!!
Worksheet Answers
1.
i 11
2. 10i
10. 17i
11.
8i 2
19.
20.
2  23i
41
3  4i
2
28. 0
29. 1
3.
2i 2
12. 6
21. 17-6i
30. 1
4.
-20
13. -5
22. 50
31. x2 - 36
5.
-8
14. 5i 2
23.  1  4i 3
32. x2 + 36
6.
i
15. -8 + 6i 24. -3 + 4i
33. x2 - 4
7. 15i
16. 2/3
25. 1 + 21i
34. x2 + 4
8. -35
17. -4i
26. -45 + 30i
35. (x+3)(x-3)
27. 1
36. (x+3i)(x-3i)
9. -36
18.
8  4i
5
37. (x+7)(x-7)
38. (x+7i)(x-7i)
Ch 5 Review Answers
3
5 6
1. 2 a b c
6b
4. 7 x2 3 y
2z
7. x  2x 7  7
10. 8  4 5
8.
3. 6 2  3
3
5. 80 3
2
2
3
2.
x 3xyz
6. 6 2  15
x2 4 12y3
5 7
9.
7
2y 2
11. 6  3 3  2 6  3 2
13. 2x 3x
1
3
 20
14. 24 3 4
15. 2
27
17. x  6
18. x  100 19. x  1,
8
22. x  6 23. x  517 24. x  28,  26
26. x  4
27. Ø
12. 3x  5  14 x  1
16. x 
5
3
20. y  1
25. x  13
21. x  9
28. a. y  x
30. a. y  2 x
29. a. y  x  2
b. y  x  2
b. y  21 x
b. y  x  3
c. y  x  2
b
a
c
a
b
b
a