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Common Core Algebra 2

Chapter 5: Rational Exponents &

Radical Functions

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Chapter Summary

• This first part of this chapter introduces radicals and 𝑛th roots and how these may be written as

rational exponents. A connection is made to the properties of exponents studied in Algebra I, noting

that now exponents can be rational numbers and are no longer restricted to being nonzero integers.

• In the middle portion of this chapter, radical expressions, also written in rational exponent form, are

presented as functions and are graphed. This leads to a look at what the domains are for each function

type.

• The graphs of radical functions are used to help students think about solutions of radical equations and

inequalities. Certainly, one goal is for students to recognize that solving radical equations is an

extension of solving other types of functions. The difference, however, is that sometimes extraneous

solutions are introduced when solving radical equations, so it is necessary to check apparent solutions.

• The last lessons in the chapter involve performing the four basic operations on function and doing so

from multiple approaches: symbolic, numerical, and graphical. The last lesson introduces inverse

functions—finding the inverse of linear, simple polynomial, and radical functions, and noting that the

graphs of inverse functions are reflections in the line 𝑦 = 𝑥.

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Section 5.1 – 𝒏th Roots and Rational Exponents

Essential Question: How can you use a rational exponent to represent a power involving a radical?

What You Will Learn ➢ Find 𝑛th roots of numbers.

➢ Evaluate expressions with rational exponents.

➢ Solve equations using 𝑛th roots.

----------------------------------------------------------------------------------------------------------------------------- -----------------------------------

Previously, you learned that the 𝑛th root of 𝑎 can be represented as:

√𝑎𝑛 = _______

for any real number 𝑎 any integer 𝑛 greater than 1.

----------------------------------------------------------------------------------------------------------------------------- ----------------------------------- EXPLORATION 1: Exploring the Definition of a Rational Exponent

Use a calculator to show that each statement is true.

(a) √9 = 91/2 (b) √2 = 21/2 (c) √83

= 81/3 (d) √124

= 121/4

----------------------------------------------------------------------------------------------------------------------------- -----------------------------------

EXPLORATION 2: Writing Expressions in Rational Exponent Form

Use the definition of a rational exponent and the properties of exponents to write each expression as a base with a

single rational exponent. Then use a calculator to evaluate each expression. Round your answers to two decimal places.

Example: (√43

)2

= (41/3)2

= 42/3

≈ 2.52

(a) (√5)3

(b) (√44

)2

(c) (√93

)2

(d) √923 (e) (√10

5)

4

_____________________

_____________________

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EXPLORATION 3: Writing Expressions in Radical Form

Use the properties of exponents and the definition of a rational exponent to write each expression as a radical raised to

an exponent. Then use a calculator to evaluate each expression. Round your answer to two decimal places.

Example: 52/3 = (51/3)2

= (√53

)2

≈ 2.92

(a) 82/3 (b) 65/2 (c) 123/4 (d) 103/2

----------------------------------------------------------------------------------------------------------------------------- -----------------------------------

COMMUNICATE YOUR ANSWER

How can you use a rational exponent to represent a power involving a radical?

Example 1: Evaluate each expression without using a calculator. Make sure you can explain your reasoning.

(a) 43/2 (b) 324/5 (c) 812/4 (d) 493/2 (e) 1006/3

**In general, for an integer 𝑛 greater than 1, if 𝑏𝑛 = 𝑎, then if we solve this equation for 𝑏, _______________________

we can say that ____________________________________________________________________________________.

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Rational Exponents

A rational exponent does not have to be of the form 1

𝑛 . Other rational numbers, such as

3

2 and −

1

2, can also be used as

exponents. Two properties of rational exponents are shown below.

Example 2: Evaluate each expression using the above property.

(a) (−64)2/3 (b) 32−3/5 (c) (−225)−1/2

Solving Equations Using 𝒏th Roots

To solve an equation of the form 𝑢𝑛 = 𝑑, where 𝑢 is an algebraic expression, take the 𝑛th root of each side.

Example 3: Find the real solution(s) of each of the following equations.

(a) 4𝑥5 = 128 (b) 1

2𝑥5 = 512 (c) (𝑥 − 3)4 = 21

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Section 5.2 – Properties of Rational Exponents and Radicals

Essential Question: How can you use properties of exponents to simplify products and quotients of radicals?

What You Will Learn ➢ Use properties of rational exponents to simplify expressions with rational exponents.

➢ Use properties of radical to simplify and write radical expressions in simplest form. -------------------------------------------------------------------------------------------------------------------------------------------------- -------------- EXPLORATION 1: Let 𝑎 and 𝑏 be real numbers. Use the properties of exponents to complete each statement. Then match each completed statement with the property it illustrates.

Statement Property

(a) 𝑎−2 = ______, 𝑎 ≠ 0 A. Product of Powers (b) (𝑎𝑏)4 = ________ B. Power of a Power (c) (𝑎3)4 = ________ C. Power of a Product (d) 𝑎3 • 𝑎4 = ________ D. Negative Exponent

(e) (𝑎

𝑏)

3= _______, 𝑏 ≠ 0 E. Zero Exponent

(f) 𝑎6

𝑎2 = _______, 𝑎 ≠ 0 F. Quotient of Powers

(g) 𝑎0 = _______, 𝑎 ≠ 0 G. Power of a Quotient

---------------------------------------------------------------------------------------------------------------------------------------------- ------------------ PROPERTIES OF RATIONAL EXPONENTS

The properties of integer exponents can also be applied to rational exponents.

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Example 1: Apply the properties of integer exponents to rational exponents by simplifying each expression. Use a calculator to check your answers.

(a) 52/3 • 54/3 (b) (101/2)4 (c)

85/2

81/2

(d) 3

31/4 (e) (51/3 • 72/3)3 (f) (

201/2

51/2 )3

----------------------------------------------------------------------------------------------------------------------------- ----------------------------------- SIMPLIFYING RADICAL EXPRESSIONS

The Power of a Product and Power of a Quotient properties can be expression using radical notation when 𝑚 =1

𝑛 for

some integer 𝑛 greater than 1.

Example 2: Use the properties of radicals to simplify each expression.

(a) √123

• √183

(b) √804

√54 (c) √135

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(d) √2884

(e) √645

(f) √3,000,0006

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SIMPLIFYING VARIABLE EXPRESSIONS The properties of rational exponents and radicals can also be applied to expressions involving variables. Because a variable can be positive, negative, or zero, sometimes absolute value is needed when simplifying a variable expression.

**Absolute value is not needed when all variables are assumed to be positive.** ----------------------------------------------------------------------------------------------------------------------------------------------------------------

Let’s prove this by graphing the function 𝑦 = √𝑥𝑛𝑛 for values of 𝑛 that are even and odd.

ODD values of 𝒏 vs EVEN values of 𝒏

𝑦 = √𝑥33 𝑎𝑛𝑑 𝑦 = √𝑥55

𝑦 = √𝑥2 𝑎𝑛𝑑 𝑦 = √𝑥44

simplifies to _____________________ simplifies to ______________________

Example 3: Simplify the following expressions.

(a) √𝑥3 (b) √𝑥4 (c) √𝑥5 (d) √𝑥6 (e) √𝑥7

(f) √64𝑥2𝑦 (g) √80𝑥4𝑦2 (h) √27𝑦63 (i) √

𝑥4

𝑦8

4 (j) √4𝑎8𝑏14𝑐55

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Section 5.3 – Graphing Radical Functions

Essential Question: How can you identify the domain and range of a radical function?

What You Will Learn ➢ Graph radical functions.

➢ Write transformations of radical functions.

➢ Graph parabolas and circles. ------------------------------------------------------------------------------------------------------------------------------------------------ ---------------- MOTIVATION

Have you ever been in a car that skidded on a road surface or have ever seen skid marks on a road? By measuring the skid marks of a vehicle and taking into account information about the efficiency of the brakes and the

surface on which the car was traveling, a police officer can use a formula involving a square root function to estimate

how fast a car was traveling at the time of an accident.

The function is 𝑆 = √30𝐷 • 𝑓 • 𝑛, where 𝑆 is the speed in miles per hour, 𝐷 is the skid distance in decimal feet, 𝑓 is the

drag factor for the road surface, and 𝑛 is the percent braking efficiency written as a decimal.

Evaluate the speed of a car if 𝐷 = 120 feet, 𝑓 = 0.75 (for asphalt), and 𝑛 = 100% (all four wheels braking).

----------------------------------------------------------------------------------------------------------------------------- -----------------------------------

EXPLORATION 1: Identifying Graphs of Radical Functions

Match each function with its graph. Explain your reasoning. Then identify the domain and range of each function.

Functions Graphs

(a) 𝑓(𝑥) = √𝑥

(b) 𝑓(𝑥) = √𝑥3

(c) 𝑓(𝑥) = √𝑥4

(d) 𝑓(𝑥) = √𝑥5

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EXPLORATION 2: Identifying Graphs of Transformations

Match each transformation of 𝑓(𝑥) = √𝑥 with its graph. Explain your reasoning. Then identify the domain and range of

each function.

Functions Graphs

(a) 𝑔(𝑥) = √𝑥 + 2

(b) 𝑔(𝑥) = √𝑥 − 2

(c) 𝑔(𝑥) = √𝑥 + 2 − 2

(d) 𝑔(𝑥) = −√𝑥 + 2

----------------------------------------------------------------------------------------------------------------------------- ----------------------------------- Communicate Your Answer

How can you identify the domain and range of a radical function?

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Graphing Radical Functions

A radical function contains a radical expression with the independent variable in the radicand. When the radical is a

square root, the function is called a square root function. When the radical is a cube root, the function is called a cube

root function.

Example 1: Graph each function. Identify the domain and range of each function.

(a) 𝑓(𝑥) = √1

4𝑥 (b) 𝑔(𝑥) = −3√𝑥3

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Transforming Radical Functions

Example 2: Describe the transformation of 𝑓 represented by 𝑔. Write the equation of 𝑔. Then sketch the graph of

𝑔 and identify the domain and range of each function.

(a) 𝑓(𝑥) = √𝑥 ; 𝑔(𝑥) = 𝑓(𝑥 − 3) + 4 (b) 𝑓(𝑥) = √𝑥3 ; 𝑔(𝑥) = 8𝑓(−𝑥)

Example 3: Let the graph of 𝑔 be a horizontal shrink by a factor of 1

6 followed by a translation 3 units to the left of the

graph of 𝑓(𝑥) = √𝑥3 . Write a rule for 𝑔.

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Section 5.4 – Solving Radical Equations and Inequalities

Essential Question: How can you solve a radical equation?

What You Will Learn ➢ Solve equations containing radicals and rational exponents.

➢ Solve radical inequalities

----------------------------------------------------------------------------------------------------------------------------- ----------------------------------- EXPLORATION 1: Solving Radical Equations

Match each radical equation with the graph of its related radical function. Explain your reasoning. Then use the graph

to solve the equation, if possible. Check your solutions.

Equation Graph # Solution

(a) √𝑥 − 1 − 1 = 0 ____________ ________________________________

(b) √2𝑥 + 2 − √𝑥 + 4 = 0 ____________ ________________________________

(c) √9 − 𝑥2 = 0 ____________ ________________________________

(d) √𝑥 + 2 − 𝑥 = 0 ____________ ________________________________

(e) √−𝑥 + 2 − 𝑥 = 0 ____________ ________________________________

(f) √3𝑥2 + 1 = 0 ____________ ________________________________

1. 2. 3.

4. 5. 6.

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Communicate Your Answer

How can you solve a radical equation?

Would you prefer to use a graphical or algebraic approach to solve the given equation? Explain your reasoning. Then

solve the equation.

√𝑥 + 3 − √𝑥 − 2 = 1

----------------------------------------------------------------------------------------------------------------------------- ----------------------------------- Solving Equations Equations with radicals that have variables in their radicands are called radical equations. An example of a radical

equation is 2√𝑥 + 1 = 4.

Example 1: Solving Radical Equations.

(a) Solve 2√𝑥 + 1 = 4 (b) √2𝑥 − 93

− 1 = 2

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Example 2: Solving a Real-Life Problem

In a hurricane, the mean sustained wind velocity 𝑣 (in meters per second) can be modeled by 𝑣(𝑝) = 6.3√1013 − 𝑝,

where 𝑝 is the air pressure (in millibars) at the center of the hurricane. Estimate the air pressure at the center of the

hurricane when the mean sustained wind velocity is 54.5 meters per second.

----------------------------------------------------------------------------------------------------------------------------- ----------------------------------- Solving an Equation with an Extraneous Solution

Raising each side of an equation to the same exponent may introduce solutions that are not solutions of the original

equation. These solutions are called _____________________________________________ solutions. When you use

this procedure you should always check each apparent solution in the original equation.

Example 3: Solve the following radical equations.

(a) 𝑥 + 1 = √7𝑥 + 15 (b) √𝑥 + 2 + 1 = √3 − 𝑥

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Solving an Equation with a Rational Exponent

When an equation contains a power with a rational exponent, you can solve the equation using a procedure similar to

the one for solving radical equations. In this case, you first isolate the power and then raise each side of the equation to

the reciprocal of the rational exponent.

Example 5: Solve the following equations.

(a) (2𝑥)3/4 + 2 = 10 (b) (𝑥 + 30)1/2 = 𝑥

----------------------------------------------------------------------------------------------------------------------------- ----------------------------------- Solving Radical Inequalities For the purpose of this class, we will solve radical inequalities using a graphical approach ONLY. Example 6: Solve the following radical inequalities.

(a) 3√𝑥 − 1 ≤ 12 (b) 4√𝑥 + 13

> 8

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Section 5.5 – Performing Function Operations Essential Question: How can you arithmetically combine the equations of two functions? What You Will Learn

➢ Add, subtract, multiply and divide functions. ----------------------------------------------------------------------------------------------------------------------------------------------------------------

Example 1: Adding Two Functions

Let 𝑓(𝑥) = 3√𝑥 and 𝑔(𝑥) = −10√𝑥. Find (𝑓 + 𝑔)(𝑥) and state the domain. Then evaluate the sum when 𝑥 = 4.

Example 2: Subtracting Two Functions

Let 𝑓(𝑥) = 3𝑥3 − 2𝑥2 + 5 and 𝑔(𝑥) = 𝑥3 − 3𝑥2 + 4𝑥 − 2. Find (𝑓 − 𝑔)(𝑥) and state the domain. Then evaluate the

difference when 𝑥 = −2.

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Example 3: Multiplying Two Functions

Let 𝑓(𝑥) = 𝑥2 and 𝑔(𝑥) = √𝑥. Find (𝑓𝑔)(𝑥) and state the domain. Then evaluate the product when 𝑥 = 9.

Example 4: Dividing Two Functions

Let 𝑓(𝑥) = 6𝑥 and 𝑔(𝑥) = 𝑥3/4. Find (𝑓

𝑔) (𝑥) and state the domain. Then evaluate the quotient when 𝑥 = 16.

Example 5: Performing Operations Using Technology

Let 𝑓(𝑥) = √𝑥 and 𝑔(𝑥) = √9 − 𝑥2. Use a graphing calculator to evaluate (𝑓 + 𝑔)(𝑥), (𝑓 − 𝑔)(𝑥), (𝑓𝑔)(𝑥), and

(𝑓

𝑔) (𝑥) when 𝑥 = 2. Round your answers to two decimal places.

Example 6: Solving a Real-Life Problem

For a white rhino, heart rate 𝑟 (in beats per minute) and life span 𝑠 (in minutes) are related to body mass 𝑚 (in

kilograms) by the functions: 𝑟(𝑚) = 241𝑚−0.25 and 𝑠(𝑚) = (6 × 106)𝑚0.2.

(a) Find (𝑟𝑠)(𝑚).

(b) Explain what (𝑟𝑠)(𝑚) represents.

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Section 5.6 – Inverse of a Function

Essential Question: How can you sketch the graph of the inverse of a function?

What You Will Learn ➢ Explore inverses of functions. ➢ Find and verify inverses of nonlinear functions. ➢ Solve real-life problems using inverse functions.

----------------------------------------------------------------------------------------------------------------------------- ----------------------------------- EXPLORATION 1: Graphing Functions and Their Inverses Each pair of functions are inverses of each other. Match each pair of equations with their corresponding graph. What do you notice about the graphs? Inverse Pairs Graphs (a) 𝑓(𝑥) = 4𝑥 + 3

𝑔(𝑥) =𝑥 − 3

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(b) 𝑓(𝑥) = 𝑥3 + 1

𝑔(𝑥) = √𝑥 − 13

(c) 𝑓(𝑥) = √𝑥 − 3

𝑔(𝑥) = 𝑥2 + 3, 𝑥 ≥ 0

(d) 𝑓(𝑥) =4𝑥 + 4

𝑥 + 5

𝑔(𝑥) =4 − 5𝑥

𝑥 − 4

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EXPLORATION 2: Sketching Graphs of Inverse Functions

Use the graph of 𝑓 to sketch the graph of 𝑔, the inverse function of 𝑓, on the same set of coordinate axes. Explain your

reasoning.

(a) (b)

(c) (d)

Communicate Your Answer

How can you sketch the graph of the inverse of a function?

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Exploring Inverses of Functions

Functions that undo each other are called inverse functions. The graph of an inverse function is a reflection of the graph

of the original function. The line of reflection is _______________.

Because inverse functions interchange the input and output values of the original function, the domain and range are

also interchanged.

To find the inverse of a function algebraically, switch the roles of 𝑥 and 𝑦, and then solve for 𝑦.

----------------------------------------------------------------------------------------------------------------------------- ----------------------------------- Example 1: Find the inverse of each of the following functions.

(a) 𝑓(𝑥) = 2𝑥 + 3 Original Function: 𝑓(𝑥) = 2𝑥 + 3

Inverse Function:

Graph of 𝑓 and its inverse:

(b) 𝑓(𝑥) = −𝑥 + 1 (c) 𝑓(𝑥) =1

4𝑥 − 2

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Inverses of Nonlinear Functions

In the previous examples, the inverses of the linear functions were also functions. However, inverses are not always

functions. The graphs of 𝑓(𝑥) = 𝑥2 and 𝑓(𝑥) = 𝑥3 are shown along with their reflections in the line 𝑦 = 𝑥.

When the domain of 𝑓(𝑥) = 𝑥2 is _____________________________ to only nonnegative real numbers, the inverse of

𝑓 is a function.

Therefore the inverse of 𝑓(𝑥) = 𝑥2, 𝑥 ≥ 0 is __________________.

------------------------------------------------------------------------------------------ ----------------------------------------------------------------------

You can use the graph of a function 𝑓 to determine whether the inverse of 𝑓 is a function by applying the

ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑙𝑖𝑛𝑒 𝑡𝑒𝑠𝑡.

Example 2: Determine whether the inverse of 𝑓 is a function.

(a) 𝑓(𝑥) = 𝑥3 − 1 (b) 𝑓(𝑥) = √𝑥 + 4 (c) 𝑓(𝑥) = 𝑥4 + 2 (d) 𝑓(𝑥) = 2√𝑥 − 53

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Verifying Functions Are Inverses

Let 𝑓 and 𝑓−1 be inverse functions. If 𝑓(𝑎) = 𝑏, then 𝑓−1( ) = ______. So in general,

𝑓(𝑓−1( )) = _____ and 𝑓−1(𝑓( )) = _____

Example 3: Verify that 𝑓(𝑥) = 3𝑥 − 1 and 𝑓−1(𝑥) =𝑥 + 1

3 are inverse functions.

Example 4: Determine whether the functions are inverses.

(a) 𝑓(𝑥) = 𝑥 + 5; 𝑔(𝑥) = 𝑥 − 5 (b) 𝑓(𝑥) = 8𝑥3; 𝑔(𝑥) = √2𝑥3

(c) 𝑓(𝑥) = √𝑥 + 9

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5 ; 𝑔(𝑥) = 5𝑥5 − 9 (d) 𝑓(𝑥) = 7𝑥3/2 − 4; 𝑔(𝑥) = (

𝑥 + 4

7)

3/2