unit 3 modeling with radical and rational...

1

Name:____________________________ Period: _____

Unit 3

Modeling with

Radical and Rational

Functions

3

Equivalent Forms of Exponential Expressions

Before we begin today’s lesson, how much do you remember about exponents? Use expanded form to write the rules for the exponents.

OBJECTIVE 1 Multiplying Exponential Expressions

32 ∙ 34 𝑦4 ∙ 𝑦10 123 ∙ 125

SUMMARY: 𝑎𝑚 ∙ 𝑎𝑛 = ___________

OBJECTIVE 2 Dividing Exponential Expressions (Remember: 𝑥

𝑥= 1)

36

32

𝑦10

𝑦4

125

123

SUMMARY: 𝑎𝑚

𝑎𝑛 = ___________

OBJECTIVE 3 Negative Exponential Expressions: Simplify 2 WAYS using expanded form AND the rule from

OBJECTIVE 2

32

36

𝑦4

𝑦10

123

125

SUMMARY: 1

𝑎𝑛 = ___________

OBJECTIVE 4 Exponential Expressions Raised to a Power

(36)2 (𝑦3)4 (12𝑚)5

SUMMARY: (𝑎𝑚)𝑛 = ___________ SUMMARY: (𝑎 ∙ 𝑏)𝑛 = ____________ We’ve learned how to simplify exponential expressions in the past and reviewed those just now. Next we need to use those properties to find some missing values.

4

Find the value of x in each of the following expressions.

5𝑥 ∙ 52 = 57

3−2 ∙ 3𝑥 = 32 42 3⁄ ∙ 4𝑥 = 4

(53)𝑥 = 56

(3−2)𝑥 =1

32

(4𝑥)1 2⁄ = 4

56

5𝑥= 54

3𝑥

312=

1

32

43 2⁄

4𝑥= 4

52

5𝑥= 1

(3𝑥)2 = 1 464𝑥 = 1

Find the values of x and y in each of the following expressions.

5𝑥

3𝑦= (

5

3)

2

(23

3𝑥)

−2

=36

2𝑦 (

𝑥2

5)

3

=26

5𝑦

(5 ∙ 6)2 = 5𝑥6𝑦

(2𝑥 ∙ 63)4 = 28 ∙ 6𝑦 (3𝑥42)3 = 4𝑦

These problems will have more than one correct solution pair for x and y. Find at least 3 solution pairs.

5

(5𝑥)(5𝑦) = 512 Possible Solutions

Option 1

Option 2

Option 3

(3𝑥)𝑦 = 33 Possible Solutions

Option 1

Option 2

Option 3

4𝑥

4𝑦= 1

Possible Solutions

Option 1

Option 2

Option 3

When there are so many rules to keep track of, it’s very easy to make careless mistakes. To help you guard against that, it helps to become a critical thinker. Take a look at the expanded and simplified examples below. One of them has been simplified correctly and there’s an error in the other two. Identify the correctly simplified example with a . For the incorrectly simplified examples, write the correct answer and provide suggestions so that the same mistake is not made again.

𝑥2

𝑥3= 𝑥

(4𝑥)(𝑥) = 4𝑥2

50𝑐2𝑑2

5𝑐𝑑5= 45𝑐2𝑑3

You’ve seen some of the more common mistakes that can happen when simplifying exponential expressions, and you may have made similar mistakes in the past. For each of the next rows of problems, complete one of the problems correctly and two of the problems incorrectly. For the incorrect problems, try to use errors that you think might go unnoticed if someone wasn’t paying close attention. When you finish, you’ll switch papers with two different neighbors (one for each row) so that they can check your work, find, fix, and write suggestions for how those mistakes can be avoided.

(2𝑥2𝑦3)5 (3𝑥)−2(𝑥2) −3𝑥2

𝑦6

2𝑥𝑦2

8𝑥2𝑦

3−224𝑥3𝑥 (−2𝑥𝑦)4

6

Kuta (Infinite Algebra I) More properties of Exponents

7

Simplifying Radicals & Basic Operations

We are familiar with taking square roots (√ ) or with taking cubed roots (√3

), but you may not be as familiar with the elements of a radical.

√𝑥𝑛

= 𝑟

An index in a radical tells you how many times you have to multiply the root times itself to get the radicand.

For example, in √81 = 9, 81 is the radicand, 9 is the root, and the index is 2 because you have to multiply the root by itself twice to get the radicand (9 ∙ 9 = 92 = 81). When a radical is written without an index, there is an understood index of 2.

√643

=? Radicand Index Root is _____ because ___ ∙ ___ ∙ ___ = ___3 = 64

√32𝑥55=?

Radicand Index

Root is ______ because ___ ∙ ___ ∙ ___ ∙ ___ ∙ ___ = ___5 = 32𝑥5

Yes…you can use your calculator to do this, but for some of the more simple problems, you should be able to figure them out in your head. To use your calculator Step 1: Type in the index. Step 2: Press MATH

Step 3: Choose 5: √𝑥

Step 4: Type in the radicand.

√243𝑦55

√1296𝑚4𝑛84 √144𝑣8

BE CAREFUL THAT YOUR VARIABLE ONLY STANDS FOR A POSITIVE NUMBER.

For instance, √𝑎2 = 𝑎, 𝑖𝑓 𝑎 ≥ 0 , but if a < 0, then √𝑎2 = −𝑎 (the opposite of a), since the square root sign

always indicates the positive square root. Since there is no way for us to know if a is positive or negative, we

use absolute value. So, √𝑎2 = |𝑎|

Example: √𝑎3𝑏2 = √𝑎 ∙ 𝑎2 ∙ 𝑏2 = 𝑎|𝑏|√𝑎

a cannot be negative because we would not have been able to take the square root of a3. However, b

could be negative, so use absolute value signs.

8

BUT not every problem will work out that nicely!

Try using your calculator to find an exact answer for √243

= ________________ The calculator will give us an estimation, but we can’t write down an irrational number like this exactly – it can’t be written as a fraction and the decimal never repeats or terminates. The best we can do for an exact answer is use simplest radical form. Here are some examples of how to write these in simplest radical form. See if you can come up with a method for doing this. Compare your method with your neighbor’s and be prepared to share it with the class. (Hint: do you remember how to make a factor tree?)

√𝟏𝟐 = 𝟐√𝟑

√𝟐𝟒𝟑

= 𝟐√𝟑𝟑

√𝟒𝟖𝟒

= 𝟐√𝟑𝟒

Simplifying Radicals: 1) ________________________________________________________

2) ________________________________________________________

3) ________________________________________________________ Examples:

216x x8 315x

3 8 4 96

3 404 364x

5 6332 yx

3 42381 zyx 3 275192 zyx 4 241875 zx

3 580n

10

Multiplying Radicals –

When written in radical form, it’s only possible to write two multiplied radicals as one if the index is the same. As long as this requirement is met,

1) multiply the _____________

2) multiply the _____________ 3) Simplify!

2532 283 10354

xyyx 53 2 323 1086 xyyx 5234 155 yxyx

33 2 854 xyx 4 334 5 402 yxx 5 535 3 9274 yxx

33 3 50253 yx 33 249 44 328

11

Adding & Subtracting Radicals

You’ve been combining like terms in algebraic expressions for a long time! Show your skills by simplifying the following expressions. 2𝑥 − 𝑥 + 4𝑥 = ______________________ 3𝑦 − 2𝑥 + 𝑦 − 6𝑦 = ____________________ Usually we say that like terms are those that contain the same variable expression, but they can also contain the same radical expression. When you add or subtract radicals, you can only do so if they contain the same index and radicand. Just like we don’t change the variable expression when we add or subtract, we’re not going to change the radical expression either. All we are going to do is add or subtract the coefficients. Always simplify the radical before you decide that you can’t add or subtract.

3 + 4 + 2 + 3 4 -

-

3 3 5 5 5 12 75

3x45 3x2033 1082325 33 54163

33 4 851252 aa 33 1357409 aa 33 4 27165 yy

503186 33 1654 44 4832

12

Simplify each expression.

9√3 + 2√3

5√2 + 2√3 3√7 − 7√73

3√32 + 2√50

√200 − √72 14√𝑥𝑦3 − 3√𝑥𝑦3

4√813

− 3√723

− 3√243

3√12 + 7√75 − √54

Simplify 2 – 7

A. -15√𝑥 B. -15x C. -5x

D. -5√𝑥

5√32 - 7√8

-7√11 + 3 √11

Multiply • . Simplify. A. 25

B. 5√53

C. 5√253

D. 5

A garden has width √13 and length 7√13. What is the perimeter of the garden in simplest radical form?

13

HW Operations with Radicals

14

Rational Exponents & Radicals Raising a number to the power of ½ is the same as performing a familiar operation. Let’s take a look

at the graph of 𝑦 = 𝑥1

2⁄ to discover that operation.

Step 1: Type 𝑥1

2⁄ into the y= screen on your graphing calculator. Step 2: Look at the table of values generated by this function. Verify that you have the same values as the rest of your class. (It is very easy to make a mistake when you type in the exponents here!) Step 3: Discuss with your classmates what you believe to be the relationship between the x and y values in the table. Where have you seen this relationship before? Summarize your findings in a sentence.

Step 4: Type 𝑥1

3⁄ into the y= screen on your graphing calculator. Step 5: Look at the table of values generated by this function. Verify that you have the same values as the rest of your class. (It is very easy to make a mistake when you type in the exponents here!) Step 6: Discuss with your classmates what you believe to be the relationship between the x and y values in the table. Have you seen this relationship before? Summarize your findings in a sentence. Step 7: Type 25𝑥 into the y= screen on your graphing calculator. Step 8: Adjust your table so that the values go up by ½ and begin at 0. Verify that your table contains the same values as the rest of your class. Step 9: Discuss with your classmates the pattern you see. Use the table below to help you see the pattern. (One row has been completed for you). Summarize your findings in the space beside the table.

X (exponent)

X (exponent) as a fraction with a

denominator of 2 Y1 (25x)

Rewrite Y1 as a power of 25 with fraction

exponents

Rewrite Y1 as a power of

√25

0

.5

1

1.5 3

2 125 253 2⁄ (√25)3

2

2.5

3

3.5

15

How could you use this pattern to find the value of 363/2? Check your answer in the calculator.

How could you use this pattern to find the value of 272/3? Check your answer in the calculator.

How could you use this pattern to find the value of 815/4? Check your answer in the calculator. Step 10: Generally speaking, how can you find the value of an expression containing a rational exponent. Use

the expression 𝑎𝑚/𝑛 to help you in your explanation. You try: Rewrite each of the following expressions in radical form.

y-9/8

(𝟑𝟐𝟓)𝟓

x 1.2

Now, reverse the rule you developed to change radical expressions into rational expressions.

2

3

x 3

2

)27( 4

5

)16( x

4

1

2a 2

7

4

5 2 53 )6( 75

7 4 39 27 3x

16

Earlier in this unit, you learned that when written in radical form, it’s only possible to write two multiplied radicals as one if the index is the same. However, if you convert the radical expressions into expressions with rational exponents, you CAN multiply or divide them (as you saw in your warm-up)! Give it a try Write your final answer as a simplified radical.

12√𝑦3

4√𝑦

(√𝑎23

√𝑏)

−6

(2√𝑎

4)

3∙ √𝑎3

√𝑥124∙ √𝑦−2 √64𝑥3

√512𝑥93 √625𝑥84

√𝑥27∙ √𝑥314

1

√−27𝑥93 (√𝑥 ∙ √𝑦23

)−6

How does the idea of simplifying radicals relate to the idea of rational exponents? There are several ways to approach this. Develop your own method for calculating simplest radical form of an expression without converting to radical form until the very last step!

𝑎32

𝑏64

𝑐105

𝑑253

Describe your method for simplifying radicals from rational exponents. Share your method with the class.

17

Radical and Rational Exponents (Kuta Software – Infinite Algebra 2)

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18

Graphing Square Root and Cube Root

Make a table for each function

𝑓(𝑥) = 𝑥2 𝑓(𝑥) = √𝑥 𝑓(𝑥) = 𝑥3 𝑓(𝑥) = √𝑥3

Ignore the points with decimals. What do you notice about the other points? ________________________ ________________________________________________________________________________________ These functions are _______________ of each other. By definition, this means the _____________ and the _____________ ______________. Plot the points from the tables above. This causes the graphs to have the _____________ _______________ but to be __________________ over the line ____________.

x y

0 1 2 3 4 5 6 7 8 9

x y

0 1 2 3 4 5 6 7 8 9

x y

0 1 2 3 4 5 6 7 8 9

x y

0 1 2 3 4 5 6 7 8 9

19

The Square Root Function

Reflect the function f(x) = x2 over the line y = x. Problems? _________________ We have to define the Square Root ______________ as ________________. This means that we will only use

the _________________ side of the graph.

The result: f(x) = √𝑥 Characteristics of the graph Vertex

End Behavior

Domain

Range

Symmetry

Pattern

General form of the Square Root Function

20

The Cube Root Function Reflect the function f(x) = x3 over the line y = x. Problems? _________________

The result: f(x) = √𝑥3

Characteristics of the graph

Vertex

End Behavior

Domain

Range

Symmetry

Pattern

General form of the Cube Root Function

21

Examples:

f(x) = √𝑥 − 3 f(x) = √𝑥3

+ 4

f(x) = −√𝑥3

f(x) = 2√𝑥 + 33

Sometimes the functions are not in graphing form. We may have to use some of our algebra skills to transform the equations into something we can use.

f(x) = √4𝑥 − 12 f(x) = √8𝑥 + 323

− 5

32

Radical Applications 1. Did you ever stand on a beach and wonder how far out into the ocean you could see? Or have you

wondered how close a ship has to be to spot land? In either case, the function hhd 2 can

be used to estimate the distance to the horizon (in miles) from a given height (in feet).

a. Cordelia stood on a cliff gazing out at the ocean. Her eyes were 100 ft

above the ocean. She saw a ship on the horizon. Approximately how far

was she from that ship?

b. From a plane flying at 35,000 ft, how far away is the horizon?

c. Given a distance, d, to the horizon, what altitude would allow you to see that far?

2. A weight suspended on the end of a string is a pendulum. The most common example of a

pendulum (this side of Edgar Allen Poe) is the kind found in many clocks. The regular back-and-

forth motion of the pendulum is periodic, and one such cycle of motion is called a period. The time,

in seconds, that it takes for one period is given by the radical equation g

lt 2 in which g is the

force of gravity (10 m/s2) and l is the length of the pendulum.

a. Find the period (to the nearest hundredth of a second) if the pendulum is 0.9 m long.

b. Find the period if the pendulum is 0.049 m long.

c. Solve the equation for length l.

d. How long would the pendulum be if the period were exactly 1 s?

33

3. When a car comes to a sudden stop, you can determine the skidding distance (in feet) for a

given speed (in miles per hour) using the formula xxs 52 , in which s is skidding distance

and x is speed. Calculate the speeding distance for the following speeds.

a. 55 mph

b. 65 mph

c. 75 mph

d. 40 mph

e. Given the skidding distance s, what formula would allow you to calculate the speed in miles

per hour?

f. Use the formula obtained in (e) to determine the speed of a car in miles per hour if the skid

marks were 35 ft long.

Solve each of the following applications. 4. The sum of an integer and its square root is 12. Find the integer.

5. The difference between an integer and its square root is 12. What is the integer?

6. The sum of an integer and twice its square root is 24. What is the integer?

7. The sum of an integer and 3 times its square root is 40. Find the integer.

34

Inverse Variation Investigation

Many people take long car trips for business, vacations, and sometimes just commuting to work. While driving at slower speeds can save gas, driving at faster speeds can save time. For example, a 300-mile trip takes 6 hours at 50 mph, but only 5 hours at 60 mph. Think about how driving time would change if an average speed of 50 mph decreased to 40 mph.

Suppose that your family is planning a 250 mile trip by car to visit relatives. Your average speed

could vary from as little as 20 mph to 60 mph or more. Your average speed depends on what roads you take, traffic, weather, speed limits, and the driver’s preferred pace.

1. How long will that 250 mile trip take if you average:

o 20 mph

o 40 mph

o 60 mph

2. Write a rule that gives time of this trip t as a function of the average driving speed s.

a. Use your calculator to make a table showing data (speed, time) for the 250 mile trip. Then sketch the graph of this relation below.

Speed

(mph)

10

15

20

25

30

35

40

45

50

55

60

65

Time

(hours)

Describe as accurately as possible, the pattern relating average speed and time of your trip. How is that pattern shown in the table and graph?

35

CHECKPOINT

In Common Core Math 1, you learned about direct variation. In direct variation, two variables have a constant ratio so that if one variable increases, so does the other.

Our road trip scenario is an example of inverse variation. Inverse variation is the opposite of direct variation in that values of the two variables change in an opposite manner – as one increases, the other decreases.

Notice the shape of the graph of inverse variation.

If the value of x _________________,

then y ____________________.

If x ______________________,

the y value __________________.

We say that y varies inversely as the value of x.

An inverse variation between 2 variables, y and x, is a relationship that is expressed

as: __________________________ where the variable k is called the _____________________ _________

_________________________.

36

GUIDED PRACTICE

1 a. What equation will relate a set distance d, average speed s, and driving time t

for a trip?

b. How does an increase in average speed change the expected driving time for

a fixed distance?

c. How is your answer to part b shown in the graphs of (speed, time) relations

for any fixed distance?

2. The distance between New York and Los Angeles is approximately 3000 miles.

a. How long will a trip from NY to LA take

By airplane, averaging 450 mph

By car, averaging 60 mph

By bicycle, averaging 15 mph

b. What equation gives the time t for the trip as a function of the average speed s?

c. Make a table showing the relation between speed and time for speeds of 50 to

500 mph. Make the graph on your calculator.

Speed 50 100 150 200 250 300 350 400 450 500

Time

d. Which change in speed causes the greater change in time for the trip: an increase from 50 to

100 mph or an increase from 450 to 500 mph?

37

INDEPENDENT PRACTICE

1. R varies inversely with variable T. If R is 168 when T = 24, find R when T = 30. HINT: Remember

𝑦 =𝑘

𝑥

2. The volume, V, of a gas varies inversely as the pressure, p, in a container. If the volume of a gas

is 200cc when the pressure is 1.6 liters per square centimeter, find the volume (to the nearest

tenth) when the pressure is 2.8 liters per sq centimeter.

3. In science, one theory of life expectancy states that the lifespan of mammals varies inversely to

the number of heartbeats per minute of the animal. If a gerbil's heart beats 360 times per

minute and lives an average of 3.5 years, what would be the life expectancy of a human with

an average of 72 beats per minute? Does this theory appear to hold for humans?

4. The values (9.7, 8) and (3, y) are from an inverse variation. Find the missing value and round to

the nearest hundredth.

5. A drama club is planning a bus trip to New York City to see a Broadway play. The cost per

person for the bus rental varies inversely as the number of people going on the trip. It will cost

$30 per person if 44 people go on the trip. How much will it cost per person if 60 people go on

the trip? Round your answer to the nearest cent, if necessary.

38

Joint and Combined Variation

1. If y varies jointly as x and z, and y = 12 when x = 9 and z = 3, find z when y = 6 and x = 15.

2. If a varies jointly as b and the square root of c, and a = 21 when b = 5 and c = 36, find a when b = 9 and c = 225.

3. Wind resistance varies jointly as an object’s surface area and velocity. If an object traveling at 40 mile per hour with a surface area of 25 square feet experiences a wind resistance of 225 Newtons, how fast must a car with 40 square feet of surface area travel in order to experience a wind resistance of 270 Newtons?

4. If y varies directly as x and inversely as z, and y = 22 when x = 4 and z = 6, find y when x = 10 and z = 25.

5. If p varies directly as the square of q and inversely as the square root of r, and p = 60 when q = 6 and r = 81, find p when q = 8 and r = 144.

6. The centrifugal force of an object moving in a circle varies jointly with the radius of the circular path and the mass of the object and inversely as the square of the time it takes to move about one full circle. A 6 gram object moving in a circle with a radius of 75 centimeters at a rate of 1 revolution every 3 seconds has a centrifugal force of 5000 dynes. Find the centrifugal force of a 14 gram object moving in a circle with radius 125 centimeters at a rate of 1 revolution every 2 seconds.

39

Modeling with Power Functions:

46

Graphing Rational Functions

General form of the Rational Function

1. x

y4

Steps to Graph Rational Functions

1st: Draw the asymptotes x = _______ and y =_______.

2nd: Plot points to L and R of vertical asymptote (at least two on

each side)

3rd: Draw the branches 4th: State the domain and range. Domain ________________________

Range _________________________

1. x

y3

Steps to Graph Rational Functions


2nd: Plot points to L and R of vertical asymptote (at least two on

each side)


Range _________________________

A

Asymptotes: x = _______, y = ________

47

3. 23

6

xy Steps to Graph Rational Functions


2nd: Plot points to L and R of vertical asymptote


Range _________________________

4. 12

3

xy

1st: Draw the asymptotes x = _______ and

y =_______.

2nd: Plot points to L and R of vertical asymptote


Range _________________________

51

Use symbolic reasoning to find all solutions for these equations. Illustrate each solution by a sketch of the

graphs of the functions involved, labeling key points with their coordinates.

a. 6

5xx

b. 4

0.5xx

c. 24

1.5xx

d. 7

10 xx

e. 22 2 4x x x f. 22 4 4 2x x x

g. 2 4 5 2 2x x x h. 23 3 1x x x

52

Solve each system symbolically. Then represent the solution on a graph. Label any key points.

1.

623

5

9

yx

xy

2.

4

2

yx

xy

3.

6yx

xy

4.

1

994

xy

yx

5.

1

322

xy

yx

6.

xy

xy 2

Solve each system symbolically.

7.

311

511

yx

yx

8.

211

411

yx

yx

9. Find the lengths of the legs of a right

triangle whose hypotenuse is 15 feet

and whose area is 3 square feet.

10. A small television is advertised to

have a picture with a diagonal

measure of 5 inches and a viewing

area of 12 square inches. What are

the length and width of the screen?

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Unit 3 Review

1. Simplify:

a) √12𝑥6𝑦7 b) 5 104 x y z c) 8 2 63 8x y z d) 11 4 775a b c

2. Write in exponential form

a) √𝑥3 b) √𝑥3𝑦6𝑧84 c) (3√𝑥

5)

2 d) (√2𝑥23

)5

3. Write in radical form

a) (𝑥3)2

5 b) (𝑥1

4𝑦2

5) c) 3𝑥2

5 d) (3𝑥)2

5

4. Graph. State the transformations in order. State the Domain and Range.

a) 3 2y x b) 3 1y x c) 2 1y x

5. Working backwards: Writing the equation when given a translation.

a) The parent function y x is translated 2 units to the left and one unit down.

b) The parent function y x is translated 3 units to the right.

c) The parent function y x is compressed vertically by a factor of 1

2 and then translated 2

units up.

54

6. Solve each radical equation.

a) 1 7x b) 5 7 25x

c) 8 4x x d) 10 2 3 1 14x

e) 3 1 4x f) 3 1 2 4x x

g) 5

6 32x h) 2

3(2 5) 16x

i) 3

2( 4) 8x j) 1

2(4 8) 2x x

k) 2 3 3 2x x

Write an equation of variation to represent the situation and solve for the missing information.

1. The volume V of a gas kept at a constant temperature varies inversely as the pressure p. If the pressure is 24 pounds per square inch, the volume is 15 cubic feet. What will be the volume when the pressure is 30 pounds per square inch?

2. The time to complete a project varies inversely with the number of employees. If 3 people can complete the project in 7 days, how long will it take 5 people?

3. The time needed to travel a certain distance varies inversely with the rate of speed. If it takes 8 hours to travel a certain distance at 36 miles per hour, how long will it take to travel the same distance at 60 miles per hour?

55

4. The number of revolutions made by a tire traveling over a fixed distance varies inversely to the radius of the tire. A 12-inch radius tire makes 100 revolutions to travel a certain distance. How many revolutions would a 16-inch radius tire require to travel the same distance?

5. For a fixed number of miles, the gas mileage of a car (miles/gallon) varies inversely with the number of gallons used. One year an employee driving a truck averaged 24 miles per gallon and used 750 gallons of gas. If the next year, to drive the same number of miles the employee drove a compact car averaging 39 miles per gallon, how many gallons of gas would be used?

Inverse Variation Modeling:

1. Complete the table.

Length (in.) 1 2 3 4 5 6 7 8

Width (in.)

2. Plot your data on the grid. Then, draw a line or curve that

seems to model the pattern in the data.

3. Describe the pattern of change in the width as the length increases. Is the relationship between length and width linear?

4. Write an equation that shows how the width w depends on the length l for rectangles with an area of 24 square inches.

Solve

unit 3 modeling with radical and rational...

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