Vocabulary to Review

Objective 7.1A

Vocabulary to Review

rational number [1.1A]

New Vocabulary

nth root of a

Rules to Review

Rules of Exponents [5.1B]

New Rules

Rule for Rational Exponents:

Discuss the Concepts

1. What is the nth root of a number?

2. Use exponents to explain why and why

3. What is the meaning of the exponential expression

4. Explain why 16-3/2 is not a negative number.

5. Explain why is not a real number when a is a negative number.

Concept Check

Determine which of the numbers is the largest.

1. a.

c. c

2. a. (-32)-1/5

b.(-32)2/5

c. (-32)1/5 b

3. a. 41/2∙ 43/2

b. 34/5∙ 36/5

c. 71/4∙ 73/4 a

4. a.

c. c

Optional Student Activity

1. The U.S. Census Bureau provides projections of resident populations in the United States. The function that approximately models a projection for the population of children ages 5 through 13 in the years 1997 to 2050 is f(x) = 43,000x-1/10 where x is the year, with x = 7 equal to 1997, and f(x) is the population in thousands.

a. Use the model to estimate the population of children ages 5 through 13 in 2010. Round to the nearest thousand. 31,869,000 children

b. Use the model to estimate the difference in this population group between 2000 and 2050. Round to the nearest thousand. 5,603,000 children

c. Who might be interested in statistics on the population of this age group? Answers will vary. Examples include schools and manufacturers of children’s clothing.

2. The Insurance Institute for Highway Safety has released data on the number of car accidents in which motorists of different ages are involved. The function that approximately models the data is f(x) = 6434x-4/3 where x is the age of the driver in years and f(x) is the number of crashes per 1000 licensed drivers.

a. Use the model to approximate the accident rate per 1000 drivers for 18-year-olds. Round to the nearest tenth. 136.4

b. Use the model to approximate the difference between the accident rate for 16-year-olds and the accident rate for 60-year-olds. Round to the nearest tenth. 132.2

c. Provide an explanation for the decrease in the accident rate as age increases. Answers will vary. For example, older drivers have more driving experience.

Objective 7.1B

Vocabulary to Review

nth root of a [7.1A]

New Vocabulary

radical

index

radicand

New Symbols

New Rules

Discuss the Concepts

1. Write two expressions that represent the nth root of a. For each expression, name the term that describes each part of the expression.

2. Write an exponential expression of the form Explain how to rewrite it as a radical expression.

3. Write a radical expression of the form SET. Explain how to rewrite it as an exponential expression.

Concept Check

Write the expression in exponential form. Then simplify the resulting expression.

1. (161/2)1/2,2

2. (43/2)1/3, 2

3. (32-4/5)1/4,

4. (243-4/5)1/2,

Optional Student Activity

Simplify the product by first rewriting each radical expression as an exponential expression and then multiplying the resulting expressions.

x – y

Objective 7.1C

Vocabulary to Review

square root [5.6A]

perfect square [5.6A]

perfect cube [5.6B]

New Vocabulary

principal square root

Discuss the Concepts

1. Which of the following represent perfect squares? Why?

2. Which of the following represent perfect cubes? Why?

3. Explain how to determine whether a radical expression is the root of a perfect power.

4. Explain why. Because (x + y)3 ≠ x3 + y3

Concept Check

1. Evaluate the expression for the given values of the variables.

a. where and -24

b. where and -30

c. where and 40

d. where and 49

e. where and 39

f. where and 23

g. , where and 5

h. where and 8

2. If is an even integer, what is a possible value of x?

Answers will vary. For example, 216, 1728, or 5832.

3. If , what is the valueof 32

Optional Student Activity

1. If means and means what is the value of the expression ? 16

2. For how many real numbers x is the expression a real number? One

3.By what factor must you multiply a number in order to double its square root? To triple its square root? To double its cube root? To triple its cube root? 4; 9; 8; 27

Answers to Writing Exercises

139. No. If x≥ 0, the statement is true. However, if x 0, then . For example, if x = -2 then , not -2.

Objective 7.2A

Vocabulary to Review

irrational number [1.1A]

perfect square [5.6A]

perfect cube [5.6B]

New Properties

The Product Property of Radicals

Discuss the Concepts

1. Which of the following represent irrational numbers?

2. Is the radical expression in simplest form?Why or why not?

3. Explain how to write in simplest form.

4. Explain how we use the Product Property of Radicals when writing a radical expression in simplest form.

Concept Check

Determine whether the statement is always true, sometimes true, or never true.

1. The fourth root of a positive number is a positive number, and the fourth root of a negative number is a negative number. Never true

2. Every positive number has two cube roots, one of which is the opposite of the other. Never true

3. The square root of a number that is not a perfect square is an irrational number. Always true

4. If the radicand of a radical expression is evenly divisible by a perfect square greater than 1, then the radical expression is not in simplest form. Sometimes true

5. If a and b are real numbers, then . Sometimes true

Objective 7.2B

Vocabulary to Review

radicand [7.1B]

index [7.1B]

Properties to Review

Distributive Property [1.3A]

Discuss the Concepts

1. Why must radical expressions have the same index and the same radicand before they can be added or subtracted?

2. Which of the following expressions cannot be simplified?

Concept Check

Given write in simplest form.

Optional Student Activity

1. Write a paragraph that compares adding two monomials to adding two radical expressions. For example, compare the addition of to the addition of .

2. Write a paragraph that compares simplifying a variable expression such as to simplifying a radical expressionsuch as .

Objective 7.2C

Vocabulary to Review

The FOIL method [5.3B]

New Vocabulary

conjugates

Properties to Review

The Product Property of Radicals [7.2A]

The Distributive Property [1.3A]

Discuss the Concepts

1. Why is it not necessary for two radical expressions that are to be multiplied to have the same radicand?

2. Must two radical expressions have the same index if they are to be multiplied? Why or why not?

3. How do you determine the conjugate of ?

4. Why does the product of two conjugates involving square roots produce an expression without a radical?

Concept Check

1. Rectangle ABCD in the rectangular coordinate system has a length of units and a width of units. Find the area of the rectangle. 40 square units

2. Verify that is asquare root of .

Optional Student Activity

1. Factor each expression over the set of real numbers. For example, since , can be factored as.

2. Simplify:

Objective 7.2D

Vocabulary to Review

conjugates [7.2C]

New Vocabulary

rationalizing the denominator

New Properties

The Quotient Property of Radicals

Discuss the Concepts

1. Must two radical expressions have the same index if they are to be divided? Why or why not?

2. Is the expression in simplest form? Why or why not?

3. Why can we multiply by without changing thevalue of the expression?

Concept Check

Simplify each of the following expressions. Then write a rule for simplifying .

The rule is.

Optional Student Activity

1. Simplify:

2. Simplify:

Answers to Writing Exercises

74. A radical expression is in simplest form when:

(1) The radicand contains no factor greater than 1 that is a perfect power of the index.

(2) There is no fraction under the radical sign.

(3) There is no radical in the denominator of a fraction.

75. To rationalize the denominator of a radical expression means to rewrite the expression with no radicals in the denominator. It is accomplished by multiplying both the numerator and denominator by the same expression, one that removes the radical(s) from the denominator of the original expression.

Objective 7.3A

New Vocabulary

radical equation

extraneous solution

New Properties

Property of Raising Each Side ofan Equation to a Power:If , then .

Discuss the Concepts

1. What is the first step in solving ? Why?

2. Why is the first step in solving the equation not to square each side of the equation?

3. What does the Property of Raising Each Side of an Equation to a Power state?

4. When both sides of an equation are raised to an even power, why is it necessary to check the solutions?

5. Suppose you solve the equation and the result is . Describe how to check the solution.

Concept Check

Determine whether the statement is always true, sometimes true, or never true.

1. We can square both sides of an equation without changing the solutions of the equation.

Sometimes true

2. The Property of Raising Each Side of an Equation to a Power is used to eliminate a radical expression from an equation. Always true

3. If , then Sometimes true

4. When you raise both sides of an equation to an even power, the resulting equation has a solution that is not a solution of the original equation. Sometimes true

5. The first step in solving a radical equation is to square both sides of the equation. Sometimes true

Optional Student Activity

1. Explain how solving theequation is similar to solving the equation

2. The equation can be used to predict the maximum speed s (in feet per second) of n rowers on a scull.

a. How many rowers are needed to travel at a speed of 35 ft/s? Round to the nearest whole number.

4 rowers

b. Does doubling the number of rowers double the maximum speed of the scull? No

Objective 7.3B

Vocabulary to Review

right triangle [3.1B]

hypotenuse [3.1B]

legs of a right triangle [3.1B]

Pythagorean Theorem [3.1B]

Concept Check

Hydroplaning occurs when, rather than gripping the road’s surface, a tire slides on the surface of water that is on the pavement.

The equation gives the relationship between v, the minimum hydroplaning speed in miles per hour, and p, the tire pressure in pounds per square inch.

1. As the tire pressure increases, does the minimum hydroplaning speed increase or decrease? How did you determine this? Increases

2. As the minimum hydroplaning speed increases, does the tire pressure increase or decrease? How did you determine this? Increases

3. Is there more danger of hydroplaning when the tire pressure is low or when the tire pressure is high? How did you determine this? When the tire pressure is low

4. What implications does this formula have for drivers with respect to checking the tires on their vehicles? Answers will vary.

Optional Student Activity

(Note: You will need a ruler and a compass for this activity.)

In this activity you will graph square roots on the number line. We will begin by explaining how to graph on the number line.

Draw a number line from -2 to 2. Leave 1 inch between each number. Starting at 0, construct triangle ABC. Leg, from 0 to 1 on the number line, is 1 unit long.Leg is perpendicular to and equal in length to. Draw from point A to point B. Triangle ABC is a right triangle. Use the Pythagorean Theorem to find the length of the hypotenuse. (The length of the hypotenuse is units.) Place the point of your compass at A (0 on the number line) and the compass pencil at point B. Draw a circle with radius AB. Label the point at which the circle intersects the number line as point D. Draw a dot at D. This is the graph of on the number line.

Use the procedure outlined above to graph and on the number line.

Objective 7.4A

Vocabulary to Review

real number [1.1A]

New Vocabulary

imaginary number

complex number

real part of a complex number

imaginary part of acomplex number

New Symbols

Discuss the Concepts

1. What does the variable i represent?

2. What is a complex number?

3. a. What is the real part of ?

b. What is the imaginary part of ?

Optional Student Activity

Find a complex number z such that 3z = -10 + 4iz. Express z in the form

Objective 7.4B

Vocabulary to Review

real part of a complex number [7.4A]

imaginary part of a complex number [7.4A]

Concept Check

One area in which complex numbers are applied is the field of electrical engineering. In an alternating current (AC) circuit, the impedance is the amount by which the circuit resists the flow of electricity. It is a measure of the opposition to the flow of electricity. It is measured in ohms and is described by a complex number.

The total impedance in a circuit is a function of the impedances and of the individual circuits. In a series circuit,

1. Find the total impedance in a series circuit when ohms and ohms.

12 ohms

2. Find the total impedance in a series circuit when ohms and ohms.

16 ohms

Optional Student Activity

Fractal geometry is the study of nonlinear dimensions. Fractal images are generated by substituting an initial value into a complex function, calculating the output, and then using the output as the next value to substitute into the function. The second output is then substituted into the function, and the process is repeated. This continual recycling of outputs is called iteration, and each output is called an iterate. Complex numbers are usually symbolized by the variable z, so z is used in the function below.

Let Begin with the initial value z = -2 + i. Determine the first four iterates of the function. 2 + 4i, 6 + 7i, 10 + 10i, 14 + 13i

Objective 7.4C

Vocabulary to Review

conjugate [7.2C]

Symbols to Review

i [7.4A]

i2 = -1 [7.4A]

Properties to Review

The Product Property of Radicals [7.2A]

New Rules

Discuss the Concepts

1. How do you determine the conjugate of ?

2. What is the value of ? Give an example of the use of when multiplying complex numbers.

3. Explain how to find the product of and

Concept Check

Determine whether the following statements are always true, sometimes true, or never true.

1. The product of two imaginary numbers is a real number. Always true

2. The product of two complex numbers is a real number. Sometimes true

3. The product of a complex number and its conjugate is a real number. Always true

Optional Student Activity

1. The property that can be used to factor the sum of two perfect squares over the set of complex numbers. For instance, Factor the following expressions over the set of complex numbers.

a. (y + i)(y – i)

b. (7x + 4i)(7x – 4i)

c. (3a + 8i)(3a – 8i)

2. Two complex numbers have a sum of and a difference of What is the product of the two complex numbers? -8 + 19i

Objective 7.4D

Vocabulary to Review

conjugate [7.2C]

Concept Check

Find the reciprocal of

Then show that the product of and its reciprocal is 1.

Optional Student Activity

1. Show that by simplifying .

2. Given that find

3. Use division of polynomials to find the remainder when

a. is divided by 26

b. is divided by 0

Answers to Writing Exercises

1. An imaginary number is a number whose square is a negative number. Imaginary numbers are defined in terms of i, the number whose square is -1.

A complex number is a number of the form , where a and b are real numbers and

2. All real numbers are complex numbers; they are complex numbers of the form where Not all complex numbers are real numbers; any complex number of the form ,b≠ 0 is not a real number.

Answers to Focus on Problem Solving: Another Look at Polya’s Four-Step Process

Here are two more examples of “words” for which the product of the numerical values of the letters equal 1,000,000:

PAYJAJY

DETTEY

Answers to Projects and Group Activities: Solving Radical Equations with a Graphing Calculator

1. 1.39

2. -2.531

3. 1.781

Answers to Projects and Group Activities: The Golden Rectangle

4. Answers will vary. Here are a few examples.

LeCorbusier’s United Nations building in New York City incorporates the golden rectangle. The building is L-shaped; it is the upright part that is a golden rectangle.

Many artists have painted on canvases that have the dimensions of the golden rectangle. Albrecht Dürer, Georges Seurat, Paul Signac, and Piet Mondrian are among them. But perhaps the most famous is Leonardo da Vinci, who painted the Mona Lisa. Instances of the golden rectangle in da Vinci’s Mona Lisa include the subject’s face; the upper portion of her face, bounded by the eyes; the region from the neck to just below the hands; and the region from the neckline on the dress to just below the arms.