• Evaluate:– 9

– 25• Solve:

– 푥 = 9– 푥 − 1 = 64

• Daily Agenda:– Grade assignments– 7.1 notes / assignment

Chapter 7

Power, Roots, and Radicals

Chapter 7.1

Nth Roots and Rational Exponents

• To evaluate nth roots of real numbers using both radical notation and rational exponent notation.

• To use nth roots to solve real-life problems.

• Why is 3 = 9?• What is the 9?

• Why is 4 = 64?• What is the 64?

• Why is 3 = 81?• What is the 81?

• For an integer 푛 greater than 1– If 푏 = 푎, then b is the nth root of a.

• Or 푎 = 푏• 푛 is the index

Relating Indices and Powers

• 푎 ∙ 푎 = 푎 • 푎 ∙ 푎 = 푎

• Let푛be an integer greater than 1 and let 푎 be a real number.– If 푛 is odd, then 푎 has one real nth root: 푎 = 푎 ⁄

– If 푛 is even and 푎 > 0, then 푎 has two real nth roots: ± 푎 = ±푎 ⁄

– If 푛 is even and 푎 = 0, then 푎 has one nth root: 0 = 0 ⁄ = 0

– If 푛 is even and 푎 < 0, then 푎 has no real nth roots.

• Find the indicated nth root(s) of 푎.– n = 5, a = -32

– n = 3, a = 64

– n = 4, a = 625

– n = 3, a = -27

• Let 푎 ⁄ be an nth root of a, and let m be a positive integer.

– 푎 ⁄ = 푎 ⁄ = 푎

– 푎 ⁄ = ⁄ = =

• Evaluating Expressions with Rational Exponents– 16 ⁄

– 64 ⁄

– 49 ⁄

– 16 ⁄

• Use a calculator to approximate:– 3

– 3

• Get 푥 by itself and then take the nth root of each side!

• Solve each equation.– 6푥 = 3750

– 푥 + 1 = 18

• Solve each equation.– 5푦 = 80

– 푦 − 1 = 32

• The rate 푟 at which an initial deposit 푃 will grow to a balance 퐴 in푡years with interest compounded n times a year is given by the formula

푟 = 푛⁄

− 1 . Find 푟 if 푃 = $1000,

퐴 = $2000, 푡 = 11 years, and 푛 = 12.

• A basketball has a volume of about 455.6 cubic inches. The formula for the volume of a basketball is 푉 = 4.18879푟3. Find the radius of the basketball.

• P404: 14 – 36 even, 42 – 51 by 3s, 54 – 60 even, 64

• Evaluate the expression.– 4 ⋅ 4

–

– 2– 2 ⋅ 3

• Daily Agenda– Grade Assignment– 7.2 notes / assignment

Chapter 7.2

Properties of Rational Exponents

• To use properties of rational exponents to evaluate and simplify expressions.

• To use properties of rational exponents to solve real-life problems.

• Properties from 6.1 can be applied to rational exponents

Let a and b be real numbers and let m and n be integers

Product of Powers Property 푎 ⋅ 푎 = 푎Power of a Power Property 푎 = 푎Power of a Product Property 푎푏 = 푎 푏Negative Exponent Property 푎 =

1푎

1푎 = 푎

Zero Exponent Property 푎 = 1Quotient of Powers Property 푎

푎 = 푎

Power of a Quotient Property 푎푏 =

푎푏

• Simplify.– 6 ⁄ ∙ 6 ⁄

– 27 ⁄ ∙ 6 ⁄

– 4 ∙ 2 ⁄

–

• Simplify.– 3 ⁄ ⋅ 3 ⁄

– 64 ⁄ ⋅ 8 ⁄

– 3 ⋅ 6 ⁄

–

–

• Simplify.– 25 ∙ 5

–

• Simplify.– 27 ⋅ 3

–

• Checklist for writing radicals in simplest form:– Apply properties of radicals– Remove any perfect nth powers– Rationalize any denominators

• Write in simplest form.– 64

–

– Like radicals have the same index and the same radicand.– Combine them just like variables.

• Perform the indicated operation.– 5 4 ⁄ − 3 4 ⁄

– 81 − 3

• Perform the indicated operation.– 6 3 ⁄ + 4 3 ⁄

– 625 − 5

• Because a variable can be positive, negative, or zero…– 푥 = 푥 when 푛 is odd– 푥 = 푥 when 푛 is even

• So absolute values are needed when simplifying variable expressions.

• For simplicity sake, we will assume all variables are positive!

• Simplify the expression. Assume all variables are positive.– 27푧

– 16푔 ℎ ⁄

–

–⁄

⁄

• Write the expression in simplest form. Assume all variables are positive.– 12푑 e 푓

–

• Simplify the expression. Assume all variables are positive.– 8푟 푠 푡

– 625푗 푘 ⁄

–

–⁄

• Perform the indicated operations. Assume all variables are positive.– 8 푥 − 3 푥

– 3푔ℎ ⁄ − 6푔ℎ ⁄

– 2 6푥 + 푥 6푥

• Simplify the expression. Assume all variables are positive.– 6 푠 − 2 푠 + 푠

– 4푚 푛 ⁄ − 6푚 푛 ⁄

– 3 6푦 + 2푦 6푦

• The weight W in tons of a whale as a function of length L in feet can be approximated by the mode 푊 = 0.1077퐿3/2. Approximate the weight of a humpback whale with length 49.17 feet.

• A pilot whale is about ⁄ the length of a blue whale. Is its weight also ⁄ the weight of the blue whale?

• P411: 24 – 90 x 3s, 96

• Simplify:– 4 푥 + 1– 푥 − 2

• Daily Agenda:– Grade assignment– 7.3 notes / assignment– 7.4 notes / assignment

– Quiz Monday!

Chapter 7.3

Power Functions and Function Operations

• To perform operations with function including power functions.

• To use power functions and function operations to solve real-life problems.

• Four basic operations with polynomial functions:

• Domain of h consists of the x-values in the domains of both f and g.– Domain of quotient does not include x-values when 푔(푥) = 0

Operation Definitions Example: f(x)=2x, g(x)=x+1

Addition ℎ(푥) = 푓(푥) + 푔(푥) ℎ(푥) = 2푥 + (푥 + 1) = 3푥 + 1Subtraction ℎ(푥) = 푓(푥) − 푔(푥) ℎ(푥) = 2푥 − (푥 + 1) = 푥 − 1Multiplication ℎ(푥) = 푓(푥)푔(푥) ℎ(푥) = (2푥)(푥 + 1) = 2푥2 + 2푥Division ℎ(푥) = 푓(푥)/푔(푥) ℎ(푥) =

2푥푥 + 1

• A function of the form 푦 = 푎푥푏– a is a real number– b is a rational number

• Let 푓(푥) = 3푥 ⁄ and 푔(푥) = 2푥 ⁄ . Find the following:– The sum

– The difference

– The domains

• Let 푓(푥) = 4푥 ⁄ and 푔(푥) = 푥 ⁄ . Find the following:– The product

– The quotient

– The domains

• The composition of the function 푓 with the function 푔 is:– ℎ(푥) = 푓(푔(푥))– The domain of h is the set of all x-values such that x is in

the domain of 푔 and 푔(푥)is in the domain of 푓– Need to pay attention to the order of functions when they

are composed!

• Let 푓(푥) = 2푥 and 푔(푥) = 푥2 − 1. find the following:– 푓(푔(푥))

– 푔(푓(푥))

– 푓(푓(푥))

– The domains

• Let 푓(푥) = 푥 and 푔(푥) = 푥 + 1. Find the following:– 푓(푔(푥))

– 푔(푓(푥))

– 푓(푓(푥))

– The domains

• You do an experiment on bacteria and find that the growth rate G of the bacteria can be modeled by 퐺 푡 = 82푡 . , and that the death rate D is 퐷(푡) = 10.8푡 . , where t is the time in hours. Find an expression for the number N of bacteria living at time t.

• A computer catalog offers computers at a savings of 15% off the retail price. At the end of the month, it offers an additional 10% off its own price.– Use composition of function to find the total percent

discount.

– What would be the sale price of a $899 computer?

• P418: 12 – 51 x 3s, 54 – 56

Chapter 7.4

Inverse Functions

• To find inverses of linear functions.• To find inverses of nonlinear functions.

• P421

• An Inverse Function maps the output values back to their original input values– So the range becomes the domain and the domain

becomes the range (or output to input and vice versa)– In other words, the x and y trade places– The graphs are always reflected over the line 푦 = 푥.

• Find the inverse of 푦 = −3푥 + 6.

• Function f and g are inverses of each other provided:– 푓(푔(푥)) = 푥– 푔(푓(푥)) = 푥

• The function g is denoted by 푓 , read as “f inverse”

• Verify that 푓(푥) = −3푥 + 6and 푓 (푥) = − 푥 +2are inverses.

• Find the inverse of 푦 = 푥 − 1

• Verify that this function and your answer are inverses.

• A model for a salary is 푆 = 10.50ℎ + 50, where S is the total salary (in dollars) for one week and h is the number of hours worked.– Find the inverse function for the model.

– If a person’s salary is $533, how many hours does the person work?

• A model for a telephone bill is 푇 = 0.05푚 + 29.95,where T is the total bill, and m is the number of minutes used.– Find the inverse model.

– If the total bill is $54.15, how many minutes were used?

• Find the inverse of the function 푓(푥) = 푥

• Find the inverse of the function 푓(푥) = 푥 , 푥 ≥ 0

• If no horizontal line intersects the graph of a function f more than once, then the inverse of f is itself a function.

• Consider the function 푓(푥) = 2푥 − 4. Determine whether the inverse of f is a function and then find the inverse.

• The volume of a sphere is given by 푉 = 휋푟3, where V is the volume and r is the radius. Write the inverse function that gives the radius as a function of the volume. Then determine the radius of a volleyball given that its volume is about 293 cubic inches.

• The volume of a cylinder with height 10 feet is given by 푉 = 10휋푟2, where V is the volume, and r is the radius. Write the inverse model that gives the radius as a function of the volume. Then determine the radius of a cylinder given that its volume is 1050 cubic feet.

• P426: 15 – 30 x 3s, 36 – 51 x 3s, 58, 62

• Graph 푦 = −2푥 + 3

• Daily Agenda:– Grade homework– Go over quiz…– 7.5 notes / assignment– 7.6 notes / assignment

Chapter 7.5

Graphing Square root and Cube Root Functions

• To graph square root and cube root functions.• To use square root and cube root functions to find

real-life quantities.

• Look at P431 and find tables for x = 0, 1 for the square roots and tables for x = -1, 0, 1 for the cube roots.

• Homework: P431 Activity ↑, P414: Quiz 1 and P429: Quiz 2

• Examples of radical functions:

• Activity: Investigating Graphs of Radical Functions– P431

• To graph 푦 = 푎 푥 − ℎ + 푘 or 푦 = 푎 푥 − ℎ + 푘, follow these steps.– Step 1: Sketch the graph of 푦 = 푎 푥 or 푦 = 푎 푥– Step 2: Shift the graph h units horizontally and k units

vertically.

• Describe how to obtain the graph of 푦 = 푥 − 2 +1 from the graph of 푦 = 푥.

• Graph 푦 = 2 푥 + 4 − 1

• Describe how to obtain the graph of 푦 = 푥 − 3 + 2from the graph of 푦 = 푥. Then graph.

• Graph 푦 = −2 푥 − 3 + 2.

• Graph 푦 = −2 푥 − 1 + 1.

• State the domain and range of the function in all of the examples.

• A model for the period of a simple pendulum as

measured in time units is given by 푇 = 2휋 , where

푇 is the time in seconds, 퐿 is the length of the pendulum in feet, and 푔 is 32 ft/sec2. Use a graphing calculator to graph the model. Then use the graph to estimate the period of a pendulum that is 3 feet long.

• The length of a whale can be modeled by 퐿 =21.04 푊, where 퐿 is the length in feet, and 푊 is the weight in tons. Graph the model, then use the graph to find the weight of a what that is 60 ft long.

• P434: 15 – 48 x 3s, 66

• Evaluate when 푥 = 4: 4푥 + 192− 2푥

• Daily Agenda:– Grade assignment– 7.6 notes / assignment

– Ch 7 Test on Thursday??

Chapter 7.6

Solving Radical Equations

• To solve equations that contain radicals or rational exponents.

• To use radical equations to solve real-life problems.

• Isolate the radical expression• Raise each side to the same power

• Solve 5 − 푥 = 0

• Solve 푥 + 6 = 12

• Solve 3푥 ⁄ = 243

• Solve 2푥 ⁄ = 8

• Solve 2푥 + 8 − 4 = 6

• Solve 4푥 + 28 − 3 2푥 = 0

• Solve 12 − 2푥 − 2 푥 = 0

• Extraneous or False solutions are ones that do not work

• They are introduced sometimes when raising each side to the same power

• MUST check all solutions to check for them!

• Solve 푥 + 2 = 2푥 + 28

• Solve 푥 − 3 = 4푥

• The strings of guitars and pianos are under tension. The speed v of a wave on the string depends on the force (tension) F on the string and the mass M per

unit length L according to the formula 푣 = ⋅ ⋅ 퐴.

A wave travels through a string with a mass of 0.2 kilograms at a speed of 9 meters per second. It is stretched by a force of 19.6 Newtons. Find the length of the string.

• Solve 푅 = 2.4 푥 + 3.9 + 7.2 for x if R = 19.8.

• P441: 18 – 57 x 3s, 66, 70, 74

• #57 needs graphing calculator• Leave all answers in fraction/radical form except

#57, 66

• Evaluate the expression. Express your answer to the nearest hundredth.

–

– . .

– Daily Agenda:• Grade assignment• 7.7 notes / assignment• Review• Test Friday!

Chapter 7.7

Statistics and Statistical Graphs

• To use measures of central tendency and measures of dispersion to describe data sets.

• To use box-and-whisker plots and histograms to represent data graphically.

• Statistics: numerical values used to summarize and compare sets of data

• Measures of Central Tendency:– Mean: average, 푥̅– Median: middle number when numbers are in order– Mode: the number that occurs most often

• The number of games won by teams in the Eastern Conference for the 1997-1998 regular season of the NHL is shown on the chart.

• Find the mean, median, and mode for the data set.

Eastern Conference

36, 39, 40, 34, 48, 33, 25, 30, 37, 17, 42, 40, 24

• Measures of dispersion tell us how spread out the data is.

• Two main measures of dispersion:– Range: the difference in the greatest and least numbers

– Standard Deviation: describes the difference between the mean and the data value

• The standard deviation of 푥 , 푥 , … , 푥

• 휎 = ̅ ̅ ⋯ ̅

• Find the range and standard deviation for the number of wins in the NHL Eastern Conference.

Eastern Conference

36, 39, 40, 34, 48, 33, 25, 30, 37, 17, 42, 40, 24

• A graphical way of representing data by drawing a box around the middle half of the data and extending “whiskers” to the minimum and maximum values.

• Draw a box-and-whisker plot for the NHL Easter Conference.

Eastern Conference

36, 39, 40, 34, 48, 33, 25, 30, 37, 17, 42, 40, 24

• A graphical display of data using a bar graph.– Has data grouped in intervals of equal width.– Number of data values in each interval is the frequency of

that interval.– Based off of a frequency distribution.

• Draw a histogram for the NHL Eastern Conference.Eastern Conference

36, 39, 40, 34, 48, 33, 25, 30, 37, 17, 42, 40, 24

• P450: 33, 34, 36, 37, 40, 41• Ch. 7 Review Worksheet