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04/21/23 Math 120 - KM 1
Chapter 8:Radical Expressions,
Equations, and Functions
• 8.1 Radical Expressions & Functions
• 8.2 Rational Numbers as Exponents
• 8.3 Simplifying Radical Expressions
• 8.4 Addition, Subtraction, and more Multiplication
• 8.5 More on Division of Radical Expressions
• 8.6 Solving Radical Equations
• 8.7 Applications involving Powers
and Roots
• 8.8 The Complex Numbers
04/21/23 Math 120 - KM 2
8.1 Radical Expressions& Functions
04/21/23 Math 120 - KM 3
Index
The “default” index is 2.
Radicand
Parts of a Radical
2means
8.1
04/21/23 Math 120 - KM 4
What’s the difference?
What are the square root(s)of 49?
Thesquare root(s)
of 49 are -7 and 7.
Every number (except zero) has two square roots.
8.1
04/21/23 Math 120 - KM 5
Is the radical different?
Well then,what is ?49
is theprincipal square root of 49.
49
749
8.1
04/21/23 Math 120 - KM 6
1
4
9
16
25
36
49
64
1
2
3
4
5
6
7
8
Counting with Radicals
8.1
04/21/23 Math 120 - KM 7
196
196
289
196
14
14
17
14
Simplify a few?
196 i14Not aReal
Number
8.1
04/21/23 Math 120 - KM 8
5
....236072
Exact vs Approximate
is an EXACT value.
5
is an APPROXIMATE
value.
8.1
04/21/23 Math 120 - KM 9
x)x(f The Square Root
Function
The DOMAIN is x > 0
x f(x)
0 0
1 1
4 2
9 3
8.1
04/21/23 Math 120 - KM 10
2 x)x(f
A minor change?
The DOMAIN is x > 2
x f(x)
2 0
3 1
6 2
11 3
8.1
04/21/23 Math 120 - KM 11
53 125
Random Radicals ?
43 64
24 16
34 81
25 328.1
04/21/23 Math 120 - KM 12
x527x
225x
449x
Tricky Problems? Absolutely!
x53 3125x
5 x25102 xx
6.1
04/21/23 Math 120 - KM 13
6.2 Rational Numbers as Exponents
8.2
04/21/23 Math 120 - KM 14
The Basic Idea!
mnnm
aa
n mnm
aa 8.2
04/21/23 Math 120 - KM 15
mnnm
aa
2
3
4 34
32
8
See How this Works?
8.2
04/21/23 Math 120 - KM 16
mnnm
aa
4
3
81 34 81
33
27
Let’s try another!
8.2
04/21/23 Math 120 - KM 17
mnnm
aa
3
2
8 23 822
4
Isn’t this fun?
8.2
04/21/23 Math 120 - KM 18
mnnm
aa
5
2
32
)( 25 32
1
22
1
4
1
Negative? OK!
6.2
04/21/23 Math 120 - KM 19
32
278
Try this one?
8.2
04/21/23 Math 120 - KM 20
Where We Left Off Last Class
04/21/23 Math 120 - KM 21
6 429 yx
Simplify?
6
14223 yx
6
4
6
2
6
2
3 yx
3
2
3
1
3
1
3 yx
3 23xy8.2
04/21/23 Math 120 - KM 22
yyy
3
2
6
1
Play by the Rules!
13
2
6
1
y 6
6
6
4
6
1
y
6
9
y
2
3
y8.2
04/21/23 Math 120 - KM 23
4
33
2
xy
What if?
12
9
12
8
xy
12
198xy
12 98xy
4 33 2 xy
8.2
04/21/23 Math 120 - KM 24
4
2
1
3
2
3
1
a
aa
You can do this!
4
2
1
3
2
3
1
a
4
6
3
6
4
6
2
a
4
6
5
a 6
20
a3
10
1
a
8.2
04/21/23 Math 120 - KM 25
5
2
3a
Rewrite in radical form
5 23 a
Reverse the Process?
8.2
04/21/23 Math 120 - KM 26
Rewrite in exponential form
32 xy 2
3
2yx
OK...now the other way!
8.2
04/21/23 Math 120 - KM 27
5 3 2x
Conquer This!
53
2
x
5
1
3
2
x
15
2
x8.2
04/21/23 Math 120 - KM 28
8.3 Simplifying Radical Expressions
8.3
04/21/23 Math 120 - KM 29
Simplify:
2950 yx
Assume that all expressions under radicals represent nonnegative
numbers.
25 .2
xyx 25 48.3
04/21/23 Math 120 - KM 30
Simplify:
3 81327 ba
3 2243 abba
-27
8.3
04/21/23 Math 120 - KM 31
Simplify:
5 4233096 zyx-32 .3
5 4346 32 zyyx
8.3
04/21/23 Math 120 - KM 32
3 1086565 zyxxy -8 .7
3 2322 725 zyzyxxy
3 2333 710 zyzyx
8.3
04/21/23 Math 120 - KM 33
35 2010 xx4 .5
xx 520 6
8.3
04/21/23 Math 120 - KM 34
ProductProperty
are Real
numbers
nnn baab nn banda
293 105 xyyx
11450 yx25 .2
yyx 25 528.3
04/21/23 Math 120 - KM 35
3 23 32 202 abba
3 5340 ba8 .5
3 252 bab
8.3
04/21/23 Math 120 - KM 36
34 2 23 xx
3
1
3
1
4
2
4
1
23 xx
12
4
12
4
12
6
12
3
23 xx12 4463 23 xx12 10432x
8.3
04/21/23 Math 120 - KM 37
QuotientProperty
are Real
numbers
nn banda
n
n
n
b
aba
0b
3
12
3
12
42
8.3
04/21/23 Math 120 - KM 38
3 2
3 25
2
16
ba
ba3
2
25
2
16
baba
3 38 ba32 ba
8.3
04/21/23 Math 120 - KM 39
6.4 Addition, Subtraction, and more
Multiplication
8.4
04/21/23 Math 120 - KM 40
504323 2 xx
0x
16 .2 25 .2
220212 xx
28x
8.4
04/21/23 Math 120 - KM 41
3 483 5 192815 yxyxxy
27 .3 64 .3
3 223 22 34315 yxyxyxyx
3 22 311 yxyx
8.4
04/21/23 Math 120 - KM 42
377
3777
217
8.4
04/21/23 Math 120 - KM 43
682325
68252325
1240215
4 .3
38030
8.4
04/21/23 Math 120 - KM 44
45 xx
4554 xxxx
20 xx
8.4
04/21/23 Math 120 - KM 45
3232
33323222
34
18.4
04/21/23 Math 120 - KM 46
7272 xx
72727272 xxxx
74 x
28x
8.4
04/21/23 Math 120 - KM 47
8.5 More on Division of Radical Expressions
8.5
04/21/23 Math 120 - KM 48
50
16
2
2
100
216
10
216
5
28
8.5
04/21/23 Math 120 - KM 49
5
75
5
25
35
5
35
8.5
04/21/23 Math 120 - KM 50
3 25
2
y 3 2
3 2
5
5
y
y
3 3
3 2
125
52
y
y
y
y
5
52 3 2
8.5
04/21/23 Math 120 - KM 51
25
7
25
25
45
257
1
257
1457 8.5
04/21/23 Math 120 - KM 52
32
32
32
32
32
3662
625
8.5
04/21/23 Math 120 - KM 53
8.6 Solving Radical Equations
8.6
04/21/23 Math 120 - KM 54
• ISOLATE the RADICAL
• Raise to the power of the index
• Check for extraneous solutions
• More than one radical?…
separate the radicals to
opposite
sides of the equation and
power
up!!
8.6
04/21/23 Math 120 - KM 55
An equation that contains a variable expression in a
radicand is a radical equation.
A POWER RULE for EQUATIONS
nn bathen,baIf
2277 xthen,xIf
3333 55 xthen,xIf
8.6
04/21/23 Math 120 - KM 56
094 x
94 x
814 x77x
22
8.6
04/21/23 Math 120 - KM 57
073 x
73 x
493 x52x
22
numbernegative
8.6
04/21/23 Math 120 - KM 58
01293 x
1293 x14493 x1353 x
22
45x8.6
04/21/23 Math 120 - KM 59
233 x
233 x
83 x11x
33
8.6
04/21/23 Math 120 - KM 60
112 xx
112 xx
12 xx
22
11212 xxx
122 xxx
8.6
04/21/23 Math 120 - KM 61
112 xx
12 xx 22
142 xx
0442 xx 02 2 x
2x8.6
04/21/23 Math 120 - KM 62
Be sure to check your solutions!
4623 x
55 xx
0534 x
33 16512 xx
8.6
04/21/23 Math 120 - KM 63
8.7 Applications involving Powers and
Roots
8.7
04/21/23 Math 120 - KM 64
An object is dropped from a bridge.
Find the distance the object has fallen when its speed reaches 120 ft/s. Use the equation,
where v is the speed of the object in feet per second an d is the distance
in feet.
dv 64
8.7
04/21/23 Math 120 - KM 65
An 18 foot ladder is leaning against a building. How high on the building
will the ladder reach when the base of the ladder is 6
feet from the building?
8.7
04/21/23 Math 120 - KM 66
8.8 The Complex Numbers
8.8
04/21/23 Math 120 - KM 67
The original i ...
The term “imaginary number” was coined in 1637 by Rene
Descartes Several subjects in physics
require complex numbers, such as quantum mechanics, general
relativity and fluid dynamics.
Also, complex numbers play a key role in chaos theory and in fractal
geometry.http://mathchaostheory.suite101.com/article.cfm/complex_numbers
Imaginary numbers were defined in 1572 by
Rafael Bombelli an Italian mathematician.
8.8
04/21/23 Math 120 - KM 68
More uses for i ...
Complex numbers are used extensively in physics to describe
Electromagnetic Waves and Quantum Mechanics.
http://www.jamesbrennan.org/jbrennan/139/notes/Complex%20Numbers/complex_numbers.htm
In electrical engineering complex numbers are used to
represent the phase of an alternating signal affected by inductance and capacitance.
However, the actual voltage or current at any time is still a real
number (which is calculated from the complex number).
8.8
http://www.articlesbase.com/k-12-education-articles/mathematics-in-physics-and-chemistry-893862.html
04/21/23 Math 120 - KM 69
Aerodynamics too…
The mapping function gives the velocity and pressures around the
airfoil. Knowing the pressure around the airfoil, allows the “lift”
to be determined.
6.8http://www.grc.nasa.gov/WWW/K-12/airplane/map.html
04/21/23 Math 120 - KM 70
Imaginary ... Not really!
http://en.wikipedia.org/wiki/Mandelbrot_set
8.8
“Mathematically, the Mandelbrot set can be defined as the set of complex c-values for which the orbit of 0 under iteration of the complex quadratic polynomial xn+1=xn
2 + c remains bounded.”
http://plus.maths.org/issue40/features/devaney/
04/21/23 Math 120 - KM 71
o
o The Complex Plane / Unit Circle
12 i1i
i
i
1 1
Imaginary axis
Real axis
8.8
04/21/23 Math 120 - KM 72
i
i
1 1
Imaginary axis
Real axis
Powers of “i” ?
Use the i-clock
0i
1i
2i
3i
4i
5i
6i
7i
8.8
04/21/23 Math 120 - KM 73
Use the i-clock
43i
10 iii 1
12 i
ii 3
3i ii 3
25i 1i ii 1
40i 0i
82i 2i
10 i
12 i8.8
04/21/23 Math 120 - KM 74
9
Use youri-magination!
1i
19
i3
i30
8.8
04/21/23 Math 120 - KM 75
500Try this one?
1i
1500
i 510
100 .5
i 5100510i8.8
04/21/23 Math 120 - KM 76
81121
Complex it is!
1i
i911
8.8
04/21/23 Math 120 - KM 77
1650
But not too complexfor you!1i
i425
25 .2
8.8
04/21/23 Math 120 - KM 78
( 11 + 4i ) + ( 9 + 3i )
i bet you can do this!
= 20 + 7i
8.8
04/21/23 Math 120 - KM 79
( -4 + 2i ) – ( 7 – 3i )
No problem...right?
= -11 + 5i
8.8
04/21/23 Math 120 - KM 80
( 3 + 2i ) + ( 3 – 2i )
AddComplex Conjugates?
Really?
= 6
These numbers add to 6 and multiply to
13
8.8
04/21/23 Math 120 - KM 81
1089125
Rewrite, then Simplify
ii 369325
4 .3 36 .3
i 384
ii 369325
8.8
04/21/23 Math 120 - KM 82
12 i(3i)(5i)
Multiply? i remember!
= 15i2
= -15
8.8
04/21/23 Math 120 - KM 83
12 i (-8i)(7i)
Here’s another!
= -56i2
= 56
8.8
04/21/23 Math 120 - KM 84
12 i (-6i)(-2i)
One More?
= 12i2
= -12
8.8
04/21/23 Math 120 - KM 85
12 i
( 4 + 3i)(5 – i)
FOiL ... i know you can!
= 20 – 4i + 15i – 3i2
= 23 + 11i
= 20 – 4i + 15i + 3
8.8
04/21/23 Math 120 - KM 86
12 i(4 + 3i)(4 – 3i)
Product ofComplex
Conjugates
= 16 – 9i2
= 16 + 9
= 25 These numbers multiply to 25 and add to
8
8.8
04/21/23 Math 120 - KM 87
12 i 6273
Convert to i then Distribute
iii 63273
21881 ii
239 i
i923
ii 6273
8.8
04/21/23 Math 120 - KM 88
i7
“Real-ize” the denominator!
ii
2
7
ii
i70 1
7i
8.8
04/21/23 Math 120 - KM 89
ii
3516
Another Denominatorto “Realize”.
ii
2
2
3
516
iii
i3
16
3
5
3
165 i
8.8
04/21/23 Math 120 - KM 90
ii
1
53Conjugate Time!
ii
1
1
2
2
1
5533
iiii
i41
2
82 i
8.8
04/21/23 Math 120 - KM 91
ii
5
122
Another Reality Check?
ii
5
5
2
2
25
1260210
iiii
i13
29
13
11
26
5822 i
8.8
04/21/23 Math 120 - KM 92
Solution Check?
Is 1 + 2i a solution of x2 – 2x + 5 = 0 ?
0522 xx521221 2 )i()i(
542441 2 iii
542441 ii
i00 0
8.8
04/21/23 Math 120 - KM 93
Seattle Fractals
http://www.fractalarts.com/ASF/NEW.html
Amazing Seattle Fractals!Fractal Art, Screensavers, Tutorials, Software & more!
Doug Harrington
04/21/23 Math 120 - KM 94
That’s All For Now!