1/8/2016Math 120 - KM1 Chapter 8: Radical Expressions, Equations, and Functions 8.1 Radical Expressions & Functions 8.2 Rational Numbers as Exponents 8.3.

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

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8.1 Radical Expressions& Functions

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Index

The “default” index is 2.

Radicand

Parts of a Radical

2means

8.1

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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

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Is the radical different?

Well then,what is ?49

is theprincipal square root of 49.

49

749

8.1

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1

4

9

16

25

36

49

64

1

2

3

4

5

6

7

8

Counting with Radicals

8.1

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196

196

289

196

14

14

17

14

Simplify a few?

196 i14Not aReal

Number

8.1

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5

....236072

Exact vs Approximate

is an EXACT value.

5

is an APPROXIMATE

value.

8.1

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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

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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

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53 125

Random Radicals ?

43 64

24 16

34 81

25 328.1

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x527x

225x

449x

Tricky Problems? Absolutely!

x53 3125x

5 x25102 xx

6.1

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6.2 Rational Numbers as Exponents

8.2

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The Basic Idea!

mnnm

aa

n mnm

aa 8.2

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mnnm

aa

2

3

4 34

32

8

See How this Works?

8.2

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mnnm

aa

4

3

81 34 81

33

27

Let’s try another!

8.2

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mnnm

aa

3

2

8 23 822

4

Isn’t this fun?

8.2

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mnnm

aa

5

2

32

)( 25 32

1

22

1

4

1

Negative? OK!

6.2

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32

278

Try this one?

8.2

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Where We Left Off Last Class

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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

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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

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4

33

2

xy

What if?

12

9

12

8

xy

12

198xy

12 98xy

4 33 2 xy

8.2

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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

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5

2

3a

Rewrite in radical form

5 23 a

Reverse the Process?

8.2

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Rewrite in exponential form

32 xy 2

3

2yx

OK...now the other way!

8.2

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5 3 2x

Conquer This!

53

2

x

5

1

3

2

x

15

2

x8.2

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8.3 Simplifying Radical Expressions

8.3

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Simplify:

2950 yx

Assume that all expressions under radicals represent nonnegative

numbers.

25 .2

xyx 25 48.3

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Simplify:

3 81327 ba

3 2243 abba

-27

8.3

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Simplify:

5 4233096 zyx-32 .3

5 4346 32 zyyx

8.3

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3 1086565 zyxxy -8 .7

3 2322 725 zyzyxxy

3 2333 710 zyzyx

8.3

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35 2010 xx4 .5

xx 520 6

8.3

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ProductProperty

are Real

numbers

nnn baab nn banda

293 105 xyyx

11450 yx25 .2

yyx 25 528.3

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3 23 32 202 abba

3 5340 ba8 .5

3 252 bab

8.3

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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

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QuotientProperty

are Real

numbers

nn banda

n

n

n

b

aba

0b

3

12

3

12

42

8.3

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3 2

3 25

2

16

ba

ba3

2

25

2

16

baba

3 38 ba32 ba

8.3

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6.4 Addition, Subtraction, and more

Multiplication

8.4

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504323 2 xx

0x

16 .2 25 .2

220212 xx

28x

8.4

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3 483 5 192815 yxyxxy

27 .3 64 .3

3 223 22 34315 yxyxyxyx

3 22 311 yxyx

8.4

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377

3777

217

8.4

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682325

68252325

1240215

4 .3

38030

8.4

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45 xx

4554 xxxx

20 xx

8.4

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3232

33323222

34

18.4

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7272 xx

72727272 xxxx

74 x

28x

8.4

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8.5 More on Division of Radical Expressions

8.5

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50

16

2

2

100

216

10

216

5

28

8.5

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5

75

5

25

35

5

35

8.5

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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

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25

7

25

25

45

257

1

257

1457 8.5

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32

32

32

32

32

3662

625

8.5

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8.6 Solving Radical Equations

8.6

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• 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

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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

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094 x

94 x

814 x77x

22

8.6

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073 x

73 x

493 x52x

22

numbernegative

8.6

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01293 x

1293 x14493 x1353 x

22

45x8.6

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233 x

233 x

83 x11x

33

8.6

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112 xx

112 xx

12 xx

22

11212 xxx

122 xxx

8.6

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112 xx

12 xx 22

142 xx

0442 xx 02 2 x

2x8.6

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Be sure to check your solutions!

4623 x

55 xx

0534 x

33 16512 xx

8.6

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8.7 Applications involving Powers and

Roots

8.7

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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

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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

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8.8 The Complex Numbers

8.8

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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

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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

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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

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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/

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o

o The Complex Plane / Unit Circle

12 i1i

i

i

1 1

Imaginary axis

Real axis

8.8

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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

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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

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9

Use youri-magination!

1i

19

i3

i30

8.8

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500Try this one?

1i

1500

i 510

100 .5

i 5100510i8.8

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81121

Complex it is!

1i

i911

8.8

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1650

But not too complexfor you!1i

i425

25 .2

8.8

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( 11 + 4i ) + ( 9 + 3i )

i bet you can do this!

= 20 + 7i

8.8

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( -4 + 2i ) – ( 7 – 3i )

No problem...right?

= -11 + 5i

8.8

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( 3 + 2i ) + ( 3 – 2i )

AddComplex Conjugates?

Really?

= 6

These numbers add to 6 and multiply to

13

8.8

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1089125

Rewrite, then Simplify

ii 369325

4 .3 36 .3

i 384

ii 369325

8.8

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12 i(3i)(5i)

Multiply? i remember!

= 15i2

= -15

8.8

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12 i (-8i)(7i)

Here’s another!

= -56i2

= 56

8.8

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12 i (-6i)(-2i)

One More?

= 12i2

= -12

8.8

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12 i

( 4 + 3i)(5 – i)

FOiL ... i know you can!

= 20 – 4i + 15i – 3i2

= 23 + 11i

= 20 – 4i + 15i + 3

8.8

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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

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12 i 6273

Convert to i then Distribute

iii 63273

21881 ii

239 i

i923

ii 6273

8.8

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i7

“Real-ize” the denominator!

ii

2

7

ii

i70 1

7i

8.8

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ii

3516

Another Denominatorto “Realize”.

ii

2

2

3

516

iii

i3

16

3

5

3

165 i

8.8

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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

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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

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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!