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WARM-UPFind 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
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
Objective: To simplify rational exponents
Radicalsn radicandRoot index
even10
86
42
xx
xx
xx
What should you do if the exponent is not even?
1559 xxx
1evenxx
Simplify: All variables are positive.
105
72
yx20
yx25
Cube roots: Look for perfect cubes in the coefficient.
How can you determine if the variable is a perfect cube?
4 1542
3 20127
3 106
zyx32
zyx40
yx27
Rational Exponentsxofrootnthex thn
n1
n xx root
power
na
n a xxx:General
23 = 8 28831
3
54 = 625 562562541
4
52
32
23
43
65
32
243278
94
2566427
Now – Try some fun problems!
Remember: Root first makes the number smaller.
Can you simplify rational exponents?
Assign 5.1a: 17-39 odd, 41-58 all
5.1b Simplifying Radical Expressions
Recall the exponent properties.
xaxb = b+ax
(xa)b = abx
a-n = na
1
=x
xb
abax
x0 = 1, 0x
Objective: To simplify rational expressions using exponent properties
Simplify. All variables are positive.
32
96 )(37. yxEx
31
8
2
324
8.
yxxy
Ex
32
4
34
32
49.
x
xEx
32
432
21
810.
zyxEx
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 Answers68.
70.
72.
74.
76.
78.
94.
108.
32
101
53
zyx
154
311
a
b
61
y
531ba
2132
1
sr
ab3
40014529 ,d,.t
1034
43
xervalinttheonxx
5.2 More Rational Exponents
Multiply:
1x2x4x2.2Ex
)1x2x3(x4.1Ex
21
35
32
32
4x3x.3Ex 3
232
5x5x.4Ex 2
121
2
332
23
32
4x4x.5Ex
Objective: To continue multiplying rational exponents
Factor with Rational Exponents
Determine the smallest exponent and factor this from all terms.
21
52
32
35
xy8yx4.9Exx2x.8Ex
6xx
6xx
6xx
6xx
74
78
52
54
31
32
21
Try these:
6x13x6
4xx3
12x5x2
51
52
32
34
35
310
Last one!
Add: Don’t forget the common denominator!
21
21
3
21
1x2
1x
x
x
x
4
xx4
Can you multiplying rational exponents?
Assign 5.2: 3-69 (x3), 77, 81, 97-100
5.2 Answers
23
x
61
k
6. 12. 18.
24. 30. t - 125 36. a + 27
42. 48. 54.
60. 66. 98.
100.
xxx 101520 23 128 21
aa 96 21
tt
yyxx 164025 21
21
21
65 x 314 31 xx
32 3
13
1xx
7x47t4 51
51
41
7
x
x
5.3 Simplified Radical Form
Properties for radicals: a, b > 0
bdacdcba.3
0bforb
a.2baba.1 n
nn
bannn
Simplify each radical means:
No perfect squares left under the
No perfect cubes left under the 3
No factors in the radicand can be written as powers of the index.
No fractions under the radical
No radicals in the denominator
Objective: To write Radicals in simplest radical form
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?
:xx 2
xxxxx 3736
The first one needs absolute value symbols to insure the answer is positive
The second does not because if x was negative, it could not be under an even root.
4 2110441710 zyx8zyx183
Simplify each: Do not assume variables are positive.
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.
Type 1: Similar to section 5.1
3 2243
4 41127
zyx48.2Ex
zyx1622.1Ex
Type 2: No radicals in the denominator.
615
516
35 2
3y3x4
3y3x4
2
4
4
7
Try these:
310
84 23
72
32
5
4
xy8
x2
yzx9
1
yx5
xy8x2
xy9
x4
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 3equivalent 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
28 318
3 38
6. 12.
18. 48.
54. 60.
66. 86.
66
xx
3
1354 2
y
xy
2
103 2
z
yzxy
3
65feet
22
84733
Review 5-1 to 5-3Questions? Remember NO CALCULATOR!
5.1: Simplify radicals and rational exponents Write radical expressions with rational exponentsEvaluate rational exponentsSimplify 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 32 5 xx
Simplify:
?yxyx:Doesyx
yx
yx93aazyx272
zyx20016169
21
21
21
61
41
32
23
41
432
3
423 1492
15104
Multiply:
5x3x2x21a35a 3
152
31
31
9x24x165xx45xx2
x15x46x5x
:Factor
52
54
31
32
51
52
31
32
3
21
21
32
2x42x
3x3
x
4
x4
y3x62
124
:Simplify
42
123
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.
225325 5323
Lets try some! **You may need to simplify first!
271251831. Ex
33 x43x4.2Ex
3 33 63 3 y6yx65yx48x3.Ex
2
1
3
44. Ex
821845. xEx
Can you add and subtract radicals?
Homework: 5.4
WARM-UP
)37)(37(
)35)(35(
Simplify.
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.
bdacdcba Then simplify if possible.
152541 .Ex 348252. Ex
636323. Ex 334. xxEx
Therefore factorable!!!
2315. xEx
Now for division. Don’t forget to rationalize the denominator!!
31
46.
Ex
Multiply the numerator & denominator by the conjugate of the denominator. Then FOIL.
24
257.
Ex2
38.
xEx
Can you multiply and divide radicals?
Homework: 5.5
5.5 Answers
6. 4200 12. 18.
24.
30. 3 36. x - 22 42.
48. 54. 60.
66.
3 514105 211050
baba 42025
6
77
yx
yx
22
bababa
2
xxx
4
2511
4914777 2 xxxxx
5.6a Equations with RadicalsObjective: To solve basic radical equations
Recall: 4x – 5 = 23 Locate the variable. Undo order of operations to isolate the variable.
+5 +54x = 28
4 4x = 7
How is similar?235x4 Procedure: Locate and isolate the radical. +5 +5
4 x = 28 4 4
x = 7 How do you undo the radical?( )2 ( )2
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:
843x2:1Ex 123x24:2Ex
3x34:3Ex
Medium:
3x3x:4Ex * Isolate the radical on one side.* Square both sides (the entire side- FOIL)* Solve the quadratic. (How?)
( )2 ( )2
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:
010a2a:5Ex 43x:6Ex 3
33 7x25x:7Ex
Can you solve basic radical equations?
Assign: 5-6 to # 35
5.6b More Solving Radical Equations
Objective: To solve radical equations with radicals on both sides and identiry extaneous rootsWhat happens when you have two radicals that
you cannot combine?
25 xx * 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.
How would each change affect the graph? Give the domain and range for each.
xy:Ex 31
Domain:
Range:
22 xy:Ex 23 xy:Ex 134 xy:Ex 3 xy:OneLast
Summary:
Can you solve radical equations with radicals on both sides and identity extaneous roots?
Assign: Rest of 5.6
How can you make the root open left? Upside down?
5.6b Solutions
42. 4
48. 5, 13
54.
56. 10
58. 10,000
60. The plume would be smaller if there was a current.
2512 1x
5.6c Solving Equations with Rational Exponents
Objective: To solve equations with rational exponents and understand extraneous roots
4x4x31
3
* 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.
( )3 ( )3
8x:1Ex 4 3
x1 = 64
45x4:2Ex32
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:
2611x3:3Ex43
36x4:4Ex52
08x6x:5Ex 40x3x:6Ex
036x13x:7Ex32
34
Miscellaneous
Completely factor: x2n – 5xn + 6
Now try:
Ex: 2x4n + x2n - 6 Ex: x2n+1 - 5xn+1 + 6x
Cancel:
2
n3
x
xn
5n3
x
x
1
52
n
n
xx
Assignment: Worksheet and begin test review.Can you solve equations with rational exponents and understand extraneous roots?
5.6c Worksheet Solutions
1. 27 10. 19. 252. 16 20. 27 & -64 3. 32 11. 81 21. 49 & 254. -32 12. 1024 22. 255. 64 13. 63, -62 23. 32,768 & -326. 14. 341 24. 9 & .25
25.7. 15. 26.
8. 14 16. 81 27.17. 32 28.
9. 18. 35 & -29 29.
30.
31.
64
1
16
1
3
7
3
4
2
15
)2)(3( nn xxx)3)(3(2 3
2
xxx
)5)(12( 221 nn xnx12 nx12 nx
nx2
6nx
5.7a Introduction to Complex Numbers
Objective: To define imaginary and complex numbers and perform simple operations on each
Real Numbers (R): the set of rational and irrational numbers.
Rational (Q) : any number that can be written as a fraction
Irrational (Ir) : non-repeating, non-terminating decimals
Integers (I or Z): positive and negative Whole numbers: no fractions or decimalsWhole (W) : {0, 1, 2, 3, …} no fractions, decimals or negatives.
Natural (N) : counting numbers no decimals, fractions, negatives, 0
Complex Numbers (C) : a + bi a = real part b = imaginary part
Imaginary (Im):
negative i 1
What are imaginary numbers?
Symbol?
Value?
Square roots of negative numbers.
i
1
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: Ex2: Ex3:
Ex4: Ex5: Ex6:
4 94 27
82 25
9
55
What is the value of i 2 ?
i 1( )2 ( )2
-1 = i2
When you get an i2 , always replace it with a -1.
i = i i2 = -1 i3 = i4 = i5 = i6 = i7
= i8 =
i9 = i10 = i11 = i12 = 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
Assign: 1-24 all
Can you define imaginary and complex numbers and perform simple operations on each?
5.7a Solutions
2. 7i 20. x = -.5 y = -5/3
4. -9i 22. x = 11/4 y = -2
6. 24. x = 2/5 y = -4
8.
10. -i
12. i
14. -1
16. x = 1 y = -4
18. x = 2/3 y = -.5
34i
35i
5.7b Operations on Complex Numbers
Objective: To perform operations on complex numbersAdd/subtract:
Compare to: (4 – 3x) + (2x – 8) =
Add: (2 + 4i) + (6 – 9i) = Subtract: (5 – 3i) – (7 – 5i) =
-x - 4
Add real part to real part and imaginary part to imaginary part
Multiply: Distribute or FOIL – all answers should be in standard complex form: a + bi
Don’t forget i2 = -1Ex1: 2i(3 – 4i) = Ex2: -4i(5 + 6i) =
Ex3: (2 – 3i)(4 + 5i) = Ex4: (4 – 2i)2 =
Division:
This is similar to rationalizing the denominator with radicals.
Type 1: -or- i4
i2
5
Type 2: -or-i23
4
ii
49
812
Recall: How did we rationalize the denominator? 32
4
Use the complex conjugate to divide complex numbers. a + bi a - bi
Assign: 5.7b: 25-77 odd 87-90
Can you perform operations on complex numbers?
5.7c and Review
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
Objective: To factor and simplify using complex numbers
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?
53021 iii:Ex 50821132 iiii:Ex 30221253 iiii:Ex
Ex4: 4x – 3 + 2yi = x + 2y – 8i Ex5: 68 ii
:Ex3
46
ii
:Ex34
27
iii:Ex 658548 2239 i:Ex
iii:Ex 23510 2 0411 2 x:Solve:Ex
Worksheet Answers1. 10. 17i 19. 28. 0
2. 10i 11. 20. 29. 1
3. 12. 6 21. 17-6i 30. 1
4. -20 13. -5 22. 50 31. x2 - 36
5. -8 14. 23. 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)
9. -36 18. 27. 1 36. (x+3i)(x-3i)
37. (x+7)(x-7)
38. (x+7i)(x-7i)
11i
22i
28i
25i
5
48 i
41
232 i
2
43 i
341 i
Ch 5 Review Answers
.274x.26
13x.2526,28x.24517x.236x.22
9x.211y.208
27,1x.19100x.186x.17
35
x.162.15424.1420x3x2.13
1x145x3.122362336.11548.10
775
.9y2
y12x.877x2x.7
1526.6380.5yx7.4
326.3z2
xyz3x.2b6cba2.1
331
32
2
4 322
32
3653
Ø
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