Radical Functions
This picture
shows a surfer in a
“barrel ride”—one of
surfing's most sought-after
experiences. Given the
right conditions, a surfer
can ride inside a wave
as it breaks.
9.1
9
With Great Power . . .
Inverses of Power Functions . . . . . . . . . . . . . . . . . . . . . 663
9.2
The Root of the Matter
Radical Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675
9.3
Making Waves
Transformations of Radical Functions . . . . . . . . . . . . . . 685
9.4
Keepin’ It Real
© Carnegie Learning
Extracting Roots and Rewriting Radicals. . . . . . . . . . . . 693
9.5
Time to Operate!
Multiplying, Dividing, Adding, and
Subtracting Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . 709
9.6
Look to the Horizon
Solving Radical Equations . . . . . . . . . . . . . . . . . . . . . . . 721
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© Carnegie Learning
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With Great Power . . .
9.1
Inverses of Power Functions
Learning Goals
In this lesson, you will:
• Graph the inverses of power functions.
• Use the Vertical Line Test to determine
Key Terms
• inverse of a function
• invertible function
• Horizontal Line Test
whether an inverse relation is a function.
• Use graphs to determine whether a function
is invertible.
• Use the Horizontal Line Test to determine
whether a function is invertible.
• Graph inverses of higher-degree
power functions.
• Generalize about inverses of even- and
odd-degree power functions.
T
he word transpose means to switch two or more items. The word combines the
Latin prefix trans-, meaning “across” or “over” and ponere, meaning “to put” or
“place.” The word interchange means the same thing as transpose.
Like many words, transpose is used in different ways in different fields:
• In music, the word transpose is most often used to mean rewriting a song in a
© Carnegie Learning
different key—either higher or lower.
• In biology, a transposable element is a sequence of DNA that can move from one
location to another in a gene.
• Magicians use transposition when they make two objects appear to switch places.
Keep an eye out for the word transpose in these lessons! What different ways can you
use the word transpose?
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Problem 1 Strike That, Invert It
Recall that a power function is a polynomial function of the form P(x) 5 a​xn​ ​, where n is a
non-negative integer.
The graphs at the end of this lesson show these 6 power functions.
L(x) 5 x, Q(x) 5 ​x​2​, C(x) 5 ​x3​ ​, F(x) 5 ​x4​ ​, V(x) 5 ​x5​ ​, S(x) 5 ​x6​ ​
9
Cut out the graphs.
1. The graph of the linear function L(x) 5 x models the width
of a square as the independent quantity and the height of
the square as the dependent quantity.
width
(2)
(1)
x
(2)
width
height
(1)
height
y
What part
or parts of this graph
don’t make sense in
terms of the quantities
in this situation?
L(x) 5 x
a. Transform the cutout so that it shows the height as
the independent quantity on the horizontal axis
and the width as the dependent quantity on the vertical
axis. Then sketch the resulting graph and label the axes.
How do I
know when I’ve got
the right graph?
y
664 Resulting Graph
© Carnegie Learning
x
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b. Describe the transformations you used to transpose the independent and
dependent quantities.
9
c. Is the resulting graph a function? Explain your reasoning.
© Carnegie Learning
d. Compare the graph of L(x) 5 x to the resulting graph. Interpret both graphs in terms
of the width and height of a square.
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2. The graph of the quadratic function Q(x) 5 ​x2​ ​models the
side length of a square as the independent quantity and the
area of the square as the dependent quantity.
area
side length
(2)
(1)
area
side length
Q(x) 5 ​x2​ ​
x
x
(2)
9
(1)
a. Transform the cutout so that it shows the area as the
independent quantity on the horizontal axis and the side
length as the dependent quantity on the vertical axis.
Then sketch the resulting graph and label the axes.
y
y
What part
or parts of this
graph don’t make sense
in terms of the quantities
in this situation?
Resulting Graph
b. Describe the transformations you used to transpose the independent and
dependent quantities.
c. Is the resulting graph a function? Explain your reasoning.
© Carnegie Learning
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d. Cole used an incorrect strategy to transpose the independent and
dependent quantities.
Cole
side length
(1)
y
(2)
(1)
(2)
x
side length
(1)
area
area
side length
(2)
(2)
area
y
side length
I can rotate the graph 90° clockwise
to transpose the independent and
dependent quantities.
9
area
(1)
x
Describe why Cole’s strategy is incorrect.
What units
are used to
describe area?
© Carnegie Learning
e. Compare the graph of Q(x) 5 ​x2​ ​to the resulting graph you
sketched. Interpret both graphs in terms of the side length
and area of a square.
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3. The graph of the cubic function C(x) 5 ​x3​ ​models the side length of a cube as the
independent quantity and the volume of the cube as the dependent quantity.
a. Transform the cutout so that it shows the volume as the independent quantity on the
horizontal axis and the side length as the dependent quantity on the vertical axis.
Then sketch the resulting graph and label the axes.
9
side length
y
(1)
volume
y
side length
(1) x
x
(2)
volume
(2)
C(x) 5 ​x3​ ​
Resulting Graph
b. Describe the transformations you used to transpose the independent and
dependent quantities.
d. Compare the graph of C(x) 5 ​x3​ ​to the resulting graph. Interpret both graphs in terms
of the side length and volume of a cube.
668 © Carnegie Learning
c. Is the resulting graph a function? Explain your reasoning.
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Problem 2 Across the Line
Recall that a function f is the set of all ordered pairs (x, y), or (x, f(x)), where for every value of
x there is one and only one value of y, or f(x). The inverse of a function is the set of all
ordered pairs (y, x), or (f(x), x).
By transforming the cutouts in Problem 1, you were able to see and sketch the inverses of
the functions L(x) 5 x, Q(x) 5 ​x2​ ​, and C(x) 5 ​x3​ ​.
9
1. Deanna discovered a way to use just one reflection to transpose the independent and
dependent quantities.
Deanna
I can reflect the graph across the line y = x by folding it diagonally to switch the independent and dependent
quantities.
y
(1)
height
(1)
width
width
height
(1)
(2)
(1)
x
height
(2)
(1)
x
(2)
width
(2)
height
x
width
(2)
height
(2)
(1)
width
y
width
height
y
U
se your cutouts and Deanna’s strategy to sketch the graphs of the inverses of F(x) 5 ​x4​ ​,
V(x) 5 ​x​5​, and S(x) 5 ​x6​ ​.
© Carnegie Learning
y
x
inverse of F(x) 5 ​x4​ ​
y
y
x
inverse of V(x) 5 ​x​5​
x
inverse of S(x) 5 ​x​6​
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If the inverse of a function f is also a function, then f is an invertible function, and its inverse
is written as ​f21
​ ​(x).
2 Which of the 6 power functions that you explored are invertible
functions? Explain your reasoning.
Is there
a pattern
here?
9
3. You used the Vertical Line Test to determine whether or not the inverse of a power
function was also a function. What test could you use on the original power function to
determine if its inverse is also a function? Explain your reasoning.
Talk the Talk
The Horizontal Line Test is a visual method to determine whether a function has an inverse
that is also a function. To apply the horizontal line test, consider all the horizontal lines that
could be drawn on the graph of the function. If any of the horizontal lines intersect the graph
of the function at more than one point, then the inverse of the function is not a function.
© Carnegie Learning
1. How does the graph of a power function and the graph of its inverse demonstrate
symmetry? Explain your reasoning.
2. If a graph passes both the Horizontal Line Test and the Vertical Line Test, what can you
conclude about the graph?
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3. If a graph passes the Vertical Line Test but not the Horizontal Line Test, what can you
conclude about the graph?
9
© Carnegie Learning
4. Given any point (x, y) on a graph, use a single transformation to transform the point to
its inverse location. What do you notice?
Be prepared to share your solutions and methods.
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© Carnegie Learning
9
672 Chapter 9 Radical Functions
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L(x) 5 x
Q(x) 5 ​x2​ ​
side length
side length
(2)
(1)
x
area
C(x) 5 ​x3​ ​
F(x) 5 ​x4​ ​
independent
independent
(2)
(1)
x
(2)
V(x) 5 ​x5​ ​
S(x) 5 ​x5​ ​
independent
independent
(2)
(1)
x
(2)
x
dependent
(1)
(2)
independent
(2)
dependent
independent
(1)
dependent
y
(1)
dependent
y
© Carnegie Learning
x
(2)
(1)
dependent
side length
(2)
volume
side length
(1)
dependent
y
(1)
volume
y
9.1 Inverses of Power Functions 504368_A2_Ch09_661-736.indd 673
9
x
(2)
(1)
(2)
width
(2)
height
width
(1)
area
y
(1)
height
y
673
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(2)
width
width
(1)
(2)
(2)
side length
side length
(1)
(2)
(2)
independent
independent
(1)
(2)
x
© Carnegie Learning
x
dependent
(2)
(2)
independent
(1)
(1)
dependent
(1)
dependent
x
y
independent
dependent
(1)
volume
x
(2)
(2)
volume
independent
(1)
dependent
independent
y
674 x
y
(1)
dependent
y
(2)
area
9
x
(2)
side length
(1)
height
side length
(1)
height
y
(1)
area
y
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9.2
The Root of the Matter
Radical Functions
Learning Goals
In this lesson, you will:
• Restrict the domain of f(x) 5 ​x2​ ​to graph the
square root function.
• Determine equations for the inverses of
Key Terms
• square root function
• cube root function
• radical function
• composition of functions
power functions.
• Identify characteristics of square root
and cube root functions, such as domain
and range.
• Use composition of functions to determine
whether two functions are inverses of
each other.
• Solve real-world problems using the square
root and cube root functions.
M
The time it takes for one swing of a pendulum can
be modeled by the inverse of a power function.
g
Wire 200 feet lon
1
2
3
4
Steel
ball
ar
p
As a Foucault pendulum swings back and forth
throughout the day, the Earth’s rotation causes it to
appear to move in a circular direction. At the North
Pole, a Foucault pendulum would appear to move
clockwise during the day. At the South Pole, it
would appear to move counterclockwise.
Swing
5 hours
Ap
© Carnegie Learning
any science museums display what is known
as a Foucault pendulum. French physicist
Léon Foucault used a device like this to
demonstrate in 1851 that the Earth was rotating in
space—although it was known long before that the
Earth rotated on its axis.
en
tm
oti
on
of t
he p
endulu
m
Foucault pendulum
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Problem 1 The Square Root Function
In the previous lesson, you learned that the inverse of a power function defined by the set of
all points (x, y), or (x, f(x)) is the set of all points (y, x), or (f(x), x).
Thus, to determine the equation of the inverse of a power function, you can transpose x and
y in the equation and solve for y.
9
Determine the inverse of the power function f(x) 5 ​x2​ ​, or y 5 ​x2​ ​.
First, transpose x and y.
y 5 ​x​2​
Is the
function f (x) 5 x​ 2​ ​
invertible?
x 5 ​y2​ ​
Then, solve for y.
__
__
​√x ​ 5 √
​ ​y2​ ​ ​
__
y 5 6​√ x ​
__
The inverse of f(x) 5 ​x2​ ​is y 5 6​√x ​ .
1. Why must the symbol 6 be written in front of the radical to write the inverse of the
function f(x) 5 ​x2​ ​?
© Carnegie Learning
2. Why is the inverse of the function f(x) 5 ​x2​ ​not written with the notation ​f21
​ ​(x)?
Explain your reasoning.
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3. The table shows several coordinates of the function f(x) 5 ​x2​ ​.
a. Use the ordered pairs in the table and what you know about inverses to graph the
__
function and the inverse of the function, y 5 6​√ x ​ . Explain your reasoning.
Now the
function and its
inverse will be on one
coordinate plane. How does
each point (x, y) of the
function map to
the inverse?
y
x
f(x) 5 ​x​2​
8
23
9
6
22
4
21
1
0
0
22
1
1
24
2
4
3
9
4
9
2
0
28 26 24 22
2
4
6
8
x
26
28
b. What point or points do the two graphs have in common? Why?
The graph in Question 3 shows that every positive real number has 2 square roots—a
positive square root and a negative square root. For example, 9 has 2 square roots, because
(23​)2​ ​5 9 and ​32​ ​5 9. The two square roots of 9 are 3 and 23.
© Carnegie Learning
When you restrict the domain of the power function f(x) 5 ​x2​ ​to values greater than or equal
to 0, the inverse of the function is called the square root function and is written as:
__
​f21
​ ​(x) 5 √
​ x ​ , for x $ 0.
4. Draw dashed line segments between the plotted points on the function for the
restricted domain x $ 0 and the corresponding inverse points.
a. List the ordered pairs of the points you connected.
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b. List the ordered pairs of the points that you did not connect.
Explain why these points are not connected.
Does restricting
the domain of the
function restrict
the range of
the inverse?
9
__
5. Graph the square root function ​f 21
​ ​(x) 5 √
​ x ​ by restricting the
2
domain of f(x) 5 ​x​ ​.
y
8
6
4
2
0
28 26 24 22
2
4
6
8
x
22
24
26
28
6.
Brent
Explain why Brent’s equation is incorrect.
678 © Carnegie Learning
f​ –1​ ​(x) = ___
​  1   ​
f (x)
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7. Describe the key characteristics of each function:
__
Function: f(x) 5 ​x2​ ​, for x $ 0
Inverse function: ​f21
​ ​(x) 5 √
​ x ​
Domain:
Domain:
Range:
x-intercept(s):
x-intercept(s):
y-intercept(s):
y-intercept(s):
Keep in mind
the restrictions placed
on f (x) to
produce ​f ​ 1​(x).
2
Range:
9
__
8. Does the inverse function ​f21
​ ​(x) 5 √
​ x ​ have an asymptote?
Explain your reasoning.
Problem 2 The Cube Root Function
The cube root function is the inverse of the power function f(x) 5 ​x3​ ​and can be written as
3
​f​21​(x) 5
​ x ​
.

1. The table shows several coordinates of the function c(x) 5 ​x​3​.
a. Use these points to graph the function and the inverse of the function, ​c21
​ ​(x).
x
c(x) 5 x
22
28
21
21
0
0
1
1
1
8
y
3
8
6
4
2
0
28 26 24 22
2
4
6
8
x
22
24
26
© Carnegie Learning
28
b. Explain how you determined the coordinates for the points on the inverse of
the function.
c. What point or points do the two graphs have in common? Why?
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2. Why is the symbol 6 not written in front of the radical to write the inverse of the
function c(x) 5 ​x3​ ​?
9
3. Why do you not need to restrict the domain of the function c(x) 5 ​x3​ ​to write the inverse
with the notation ​c​21​(x)?
4. Describe the key characteristics of each function:
Function: c(x) 5 ​x3​ ​
Domain:
Range:
x-intercept(s):
x-intercept(s):
y-intercept(s):
y-intercept(s):
3
Inverse function: ​c​21​(x) 5 
​ x ​

Domain:
Range:
The inverses of power functions with exponents greater than or equal to 2, such as the
square root function and the cube root function, are called radical functions. Radical
functions are used in many areas of science, including physics and computer science.
680 © Carnegie Learning
3
5. Does the inverse function ​c​21​(x) 5 
​ x ​
 have an asymptote? Explain your reasoning.
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Problem 3 Inverse by Composition
__
You know that when the domain is restricted to x $ 0, the function f(x) 5 √
​ x ​ is the inverse of
3
the power function g(x) 5 ​x​2​. You also know that the function h(x) 5 
​ x ​
  is the inverse of the
power function q(x) 5 ​x3​ ​.
The process of evaluating one function inside of another function is called the composition
of functions. For two functions f and g, the composition of functions uses the output of g(x)
as the input of f(x). It is notated as (f + g)(x) or f(g(x)).
9
__
To write a composition of the functions g(x) 5 ​x2​ ​and f(x) 5 √
​ x ​ when the domain of g(x) is
restricted to x $ 0, substitute the value of one of the functions for the argument, x, of the
other function.
__
f(x) 5 √
​ x ​  g(x) 5 ​x2​ ​
__
f(g(x)) 5 √
​ ​x2​ ​ ​ 5 x, for x $ 0
You can write the composition of these two functions as f(g(x)) 5 x for x $ 0.
__
© Carnegie Learning
1. Determine g(f(x)) for the functions g(x) 5 ​x2​ ​and f(x) 5 √
​ x ​ for x $ 0.
If f(g(x)) 5 g(f(x)) 5 x, then f(x) and g(x) are inverse functions.
2. Are f(x) and g(x) inverse functions? Explain your reasoning.
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3. Algebraically determine whether each pair of functions are inverses. Show your work.
3
a. Verify that h(x) 5 
​ x ​
  is the inverse of q(x) 5 ​x3​ ​.
9
b. Determine if k(x) 5 2​x2​ ​1 5 and j(x) 5 22​x2​ ​2 5 are inverse functions.
?
4. Mike said that all linear functions are inverses of themselves because f(x) 5 x is the
inverse of g(x) 5 x.
Is Mike correct? Explain your reasoning.
© Carnegie Learning
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Problem 4 Pendula
The time it takes for one complete swing of a pendulum depends on the length of the
pendulum and the acceleration due to gravity.
__
√__​ gL  ​ ​,
The formula for the time it takes a pendulum to complete one swing is T 5 2p ​
where T is time in seconds, L is the length of the pendulum in meters, and g is the
acceleration due to gravity in meters per second squared.
9
1. If the acceleration due to gravity on Earth is 9.8 m/​s2​ ​, write a function T(L) that
represents the time of one pendulum swing.
2. Graph the function T(L).
Use a calculator
to determine the
approximate locations
of the points.
y
8
6
Time (seconds)
4
2
0
22
2
4
6
8
10
12
14 x
22
24
26
28
© Carnegie Learning
Length of Pendulum (meters)
3. Describe the characteristics of the function, such as its domain, range, and intercepts.
Explain your reasoning.
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4. How long does it take for one complete swing when the length of the pendulum is
0.5 meter?
9
5. A typical grandfather clock pendulum completes a full swing in 2 seconds. Use your
graph to determine the approximate length of a grandfather clock pendulum.
Talk the Talk
2. When a function has an asymptote, will its inverse have an asymptote? If so, describe
the location of the asymptote for the function’s inverse.
© Carnegie Learning
1. How can knowing the domain, range, intercepts, and other key characteristics of a
power function help you determine those characteristics for the function’s inverse?
Explain your reasoning.
Be prepared to share your solutions and methods.
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Making Waves
9.3
Transformations of Radical Functions
Learning Goals
In this lesson, you will:
• Graph transformations of radical functions.
• Analyze transformations of radical functions using transformational function form.
• Describe transformations of radical functions using algebraic, graphical, and
verbal representations.
• Generalize about the effects of transformations on power functions and their inverses.
S
ome people think that they won’t need math if they choose to work in an artistic
career. Not so! Much of the graphic and animation work you see on television, in
movies, and even in print and art galleries is done on the computer, using
sophisticated graphic design software.
© Carnegie Learning
To use many graphic design programs, a knowledge of transformations, like
reflections and rotations, coordinate systems, ratios, and on and on, is essential to
working efficiently and accurately—and to get just the right effect.
How do you think knowledge about power functions and radical functions can be
used in graphic design?
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Problem 1 Shifting Sands
Recall that transformations performed on a function f(x) to form a new function g(x) can be
described by the transformational function:
g(x) 5 Af(B(x 2 C)) 1 D
A group of art students had the idea to use transformations of
radical functions to create a logo for the Radical Surfing School.
9
__
To start, they graphed the function f(x) 5 √
​ x ​ , for 0 # x # 14, and
shifted copies of the curve to create the waves g(x), h(x), and k(x).
The square
root function has
a restricted domain.
Now the dimensions of
the logo will restrict it
even more!
y
10
5
f(x)
0
g(x)
5
h(x)
10
k(x)
x
1. Do the transformations of f(x) shown on the graph take
place inside the function or outside the function? Explain your reasoning.
3. Write the domain of each transformed function as an inequality statement using the
dimensions of the logo.
686 © Carnegie Learning
2. What value or values in the transformation function were changed to create
these curves? Explain your reasoning.
Chapter 9 Radical Functions
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?
4. Devin, Stuart, and Kristen each wrote an equation for a function that was added to the
graph first using the transformational function form of f(x), and then in terms of x.
24
• Devin’s equation: g(x) 5 f(x)
__
5√
​ x ​ 2 4
• Stuart’s equation: h(x) 5 f(x 2 8)
______
5√
​ x 2 8 ​
9
• Kristen’s equation: k(x) 5 f(x 1 12)
_______
5√
​ x 1 12 ​
a. Describe whether each student’s equation is correct or incorrect.
Explain your reasoning.
© Carnegie Learning
b. Write the correct equations to describe the 3 new functions shown in the graph first
using transformational function form of f(x), and then in terms of x. Finally, write their
domains as inequality statements.
__
f(x) 5 √
​ x ​
g(x) 5
5
h(x) 5
k(x) 5
Domain:
Domain:
5
Domain:
5
Domain:
9.3 Transformations of Radical Functions 504368_A2_Ch09_661-736.indd 687
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5. The students decide that reflecting each curve, g(x), h(x), and k(x), across the respective
lines where x 5 C will make them look more like waves crashing on the beach.
a. Graph the resulting functions f9(x), g9(x), h9(x), and k9(x). Write each function first in
terms of their transformations of f(x), g(x), h(x), and k(x), and then in terms of x.
Finally, state the domain of each.
y
10
9
5
0
5
10
x
You can
use the prime
symbol (’) to
indicate that a function
is a transformation of
another function.
c. How did the domain of each transformed function change as a result of the
reflection across x 5 C?
© Carnegie Learning
b. Describe how you used the transformation function to determine the equations of
the new functions.
d. Why does your graph show only 3 curves when the original graph had 4?
Explain your reasoning.
688 Chapter 9 Radical Functions
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6. Suppose the students wanted to reflect the 3 new waves g9(x), h9(x), and k9(x) across the
line y 5 0.
a. Describe how you can use the transformational function to determine the equations
of the reflected functions.
9
b. Write three new functions using transformational form to represent each reflection
of g9(x), h9(x), and k9(x), and then each in terms of x. Use the double prime symbol (0)
to indicate each transformed function. Finally, write the domain of each
transformed function.
7. Jamal wants to add waves below the 3 waves as
shown. These waves should be copies of g9(x), h9(x),
and k9(x), except half as high and shifted to the left
2 units.
a. Write 3 new functions q(x), r(x), and s(x) in terms of
g9(x), h9(x), and k9(x) to create the waves that Jamal
wants. Make sure to write the domains of each
transformed function.
y
10
5
g9(x)
© Carnegie Learning
0
5
h9(x)
k9(x)
10
b. Describe how you used what you know about transformational function form to
determine your answer to part (a).
9.3 Transformations of Radical Functions 504368_A2_Ch09_661-736.indd 689
x
689
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8. The art students want to add some clouds to the top of the logo. For the clouds, they
3
will use the inverses of cubic functions. They start with the function c(x) 5 2
​ x ​
 1 14.
y
10
5
9
0
5
10
x
a. Transform this function and write 2 more equations to create the clouds the students
want. Graph the results.
© Carnegie Learning
b. Color the graph to show the waves and the clouds on the logo.
690 Chapter 9 Radical Functions
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Yo
Text H
ur
e
er
In many graphic design programs, a trace path can be created. A trace
path is an invisible line or curve that acts as the baseline of text that is
added to the design. When you insert text on a trace path, the text follows
the line or curve. The text shown, for example, follows the curve f(x) 5 2​x2​ ​.
9. The art students are experimenting with different square root and cube root function
graphs to use as trace paths for the surfing school’s name: Radical Surfing School.
They have narrowed their trace paths down to 2 choices. The graphs of the functions
are shown.
9
3

h(x) 5 
​3 2(x
2 1) ​
j(x) 5 2
​ x
2 1 ​
y
8
6
4
2
0
220 215 210 25
5
10
15
20 x
22
24
26
28
a. Compare and contrast the graphs of the functions and their equations. What do
you notice?
© Carnegie Learning
b. Compare the effects of increasing the A-value with increasing the B-value in a
radical function. What do you notice?
c. Label each graph with the correct equation and include the domain restrictions.
9.3 Transformations of Radical Functions 504368_A2_Ch09_661-736.indd 691
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10. Choose one of the cube root functions as a trace path for the title of the surfing school.
Or, write a different radical function to use as a trace path. Graph the function on the
coordinate plane in Question 8, and write the title of the school on the trace path.
© Carnegie Learning
9
Be prepared to share your methods and solutions.
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Keepin’ It Real
9.4
Extracting Roots and Rewriting Radicals
Learning Goals
In this lesson, you will:
• Extract roots from radicals.
• Rewrite radicals as powers that have rational exponents.
• Rewrite powers that have rational exponents as radicals.
R
© Carnegie Learning
adicals can produce imaginary results. For example, the square root of 24 is
___
equal to 2i, √
​ 24 ​ 5 2i. But, in this chapter we are not going to talk about
imaginary numbers. We are going to keep it real!
693
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Problem 1 Root of the Problem
Previously, you have rewritten radicals by extracting roots involving numbers. In this lesson
n n
you will explore how to extract roots for expressions of the form 
​ ​x
​ ​ ​. To determine how to
extract a variable from a radical, let’s consider several different values of n.
n n
1. For each value of n for the expression 
​ ​x
​ ​ ​, complete the table and sketch the graph.
Then identify the function family associated with the graph and write the
corresponding equation.
9
a. Let n 5 2.
x
​ ​n​5 ​x2​ ​
x
y
2
2
n
​  ​x
​ ​ ​ 5 
​  ​x
​ ​ ​

n
8
6
22
4
21
2
0
28 26 24 22
1
0
2
4
6
8
x
2
4
6
8
x
22
24
2
26
28
Function family of the graph:
Equation of the graph:
b. Let n 5 3.
x
​ ​n​5 ​x3​ ​
x
y
3
3
n
​  ​x
​ ​ ​ 5 
​  ​x
​ ​ ​

n
8
6
22
4
21
1
2
28 26 24 22
0
22
24
26
© Carnegie Learning
2
0
28
694 Function family of the graph:
Equation of the graph:
Chapter 9 Radical Functions
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c. Let n 5 4.
x
​ ​n​5 ​x4​ ​
x
y
4
4
n
​  ​x
​ ​ ​ 5 
​  ​x
​ ​ ​

n
8
22
6
4
21
2
0
28 26 24 22
1
0
2
4
6
8
x
2
4
6
8
x
9
22
24
2
26
28
Function family of the graph:
Equation of the graph:
d. Let n 5 5.
x
​ ​n​5 ​x5​ ​
x
y
5
5
n
​  ​x
​ ​ ​ 5 
​  ​x
​ ​ ​

n
8
6
22
4
21
2
0
1
2
28 26 24 22
0
22
24
26
© Carnegie Learning
28
Function family of the graph:
Equation of the graph:
9.4 Extracting Roots and Rewriting Radicals 504368_A2_Ch09_661-736.indd 695
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e. Analyze your representations for each value of n. What do you notice?
9
n n
To extract a variable from a radical, the expression 
​ ​x
​ ​ ​ can be written as:
n
 |x|, x,
n
​ ​x
​ ​ ​ 5 ​

when n is even  ​​
when n is odd
7
2. Explain why 
​7 ​x
​ ​ ​ 5 |x| is incorrect, for real values of x.
?
One way
7 7
to say​ 
​
x​​ ​ is
“the seventh root of x
to the seventh.”
__
3. Asia and Melissa shared their work for extracting the root from √
​ ​x4​ ​ ​,  for real values
of x.
Melissa
​  x4 ​ 5 |x2|

Who’s correct? Explain your reasoning.
696 __
​√x4 ​ 5 x2
© Carnegie Learning
Asia
Chapter 9 Radical Functions
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Problem 2 Sort It Out
Let’s review the properties of powers.
© Carnegie Learning
1. Write an explanation for each property to complete the table.
Property of
Powers
Rule
Product of Powers
​ m​ ​? ​an​ ​5 ​am 1 n
a
​ ​
Quotient of Powers
​a​m ​​ 5 ​a​m 2 n​
​ ___
​an​ ​
Power to a Power
(​am​ ​​)​n​5 ​amn
​ ​
Product to a Power
(​am​ ​? ​bn​ ​​)​p​5 ​amp
​ ​? ​bnp
​​
Quotient to a Power
​ ​a​  ​ ​ )​​ ​5 ___
​ ​a​  ​​
(​​ ___
​b​ ​
​b​ ​
Zero Power
​ 0​ ​5 1, if a fi 0
a
Negative Exponent
In Numerator
​ ​2m​5 ___
a
​ 1m  ​ ,
​a​ ​
if a fi 0 and m . 0
Negative Exponent
In Denominator
Written Explanation
9
m p
mp
n
np
____
​  1
​ m​ ​,
​ 5 a
2m
​a​ ​
if a fi 0 and m . 0
9.4 Extracting Roots and Rewriting Radicals 504368_A2_Ch09_661-736.indd 697
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© Carnegie Learning
9
698 Chapter 9 Radical Functions
504368_A2_Ch09_661-736.indd 698
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2. Cut out the items and tape each item into the appropriate group on the next page.
​ ​0​
a
​a​6​ ​
​ __
​a6​ ​
(2​a)​4​
__
​ ​a​ ​ ​
​a​ ​? a
​a​ ​ ​
​ __
​ ​0​? a
a
​ 4​ ​
(​​ __​ ​a1​  ​ ​ )​​​
a?a
​ ​25​
(​​ ​a​ ​21 ​ ​ )8​​ ​
​ 24
a
​ ​? a
​ 0​ ​
​(​a212
​ ​)​3​
​ 24
​ ​
​ ​4​? a
a
​ ​  ​ ​​3​
​​___
​ ​b26
​a​ ​
​(a
​ ​2​)2​ ​
​  b22  ​  ​​ ​
​​___
​a​ ​
​(a
​ b​2​)2​ ​
__
​ 1  ​
3
__
​ 1  ​
9
6
​ 2​ ​
a
2
2
__
1 ​
​ __
(  )
1  ​
12 ​ __
(  )
2
​ 4​ ​
a
___
​  1  ​
​a24
​ ​
© Carnegie Learning
​(​a​8​​b​4​)​2​
3
​ 7​ ​
a
9.4 Extracting Roots and Rewriting Radicals 504368_A2_Ch09_661-736.indd 699
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© Carnegie Learning
9
700 Chapter 9 Radical Functions
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1
9
​a​ ​
4
​ 24
a
​ ​
© Carnegie Learning
​ 2​ ​​b​4​
a
​ 4​ ​​b​2​
a
9.4 Extracting Roots and Rewriting Radicals 504368_A2_Ch09_661-736.indd 701
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Problem 3 The Power of Radicals!
You can rewrite a radical as a power with a rational exponent, and rewrite a power with a
rational exponent as a radical.
__
9
Solve the equation √
​ x ​ 5 ​xa​​for a, given x $ 0, to determine the exponential
__
form of √
​ x ​ .
__
​√x ​ 5 ​xa​​
__
​(​√x ​ )​2​5 ​(​xa​​)2​ ​
Square each side of the equation.
x 5 ​x​2a​
Because the bases are the same,
you can set the exponents equal
to each other.
1 5 2a
1  ​
a 5 ​ __
2
Divide by 2 to solve for a.
The exponential form of the square root of x given x $ 0, is x to the
one-half power.
__
1
__
​√x ​ 5 ​x​ ​2  ​​, given x $ 0
2. How do you know when the initial x-value can be any real number or when the initial
x-value should be restricted to a subset of the real numbers?
702 © Carnegie Learning
1. Why was the restriction “given x $ 0” stated at the beginning of the worked example?
Chapter 9 Radical Functions
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3. Determine the power that is equal to the radical.
a. Write and solve an equation to determine the power that is equal to the cube
root of x.
9
b. Write and solve an equation to determine the power that is equal to the cube root of
x squared.
4. Complete the cells in each row. In the last column, write “x $ 0” or “all real numbers” to
describe the restrictions that result in equal terms for each row.
Radical
Form
Radical to a
Power Form
2
​4 ​x
​ ​ ​

4
(
​ x ​​
 )2​ ​
Exponential
Form
Restrictions
3
__
x​ ​ ​4 ​ ​
2
__
x​ ​ ​5  ​​
​ x ​


© Carnegie Learning
5
a
__
n a
You can rewrite a radical expression 
​ ​x
​​ ​ as an exponential expression ​x​ ​n  ​​:
• For all real values of x if the index n is odd.
• For all real values of x greater than or equal to 0 if the index n is even.
9.4 Extracting Roots and Rewriting Radicals 504368_A2_Ch09_661-736.indd 703
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Problem 4 Extracting Roots and Rewriting Radicals
You can extract roots to rewrite radicals, using radicals or powers.
3

Extract the roots and rewrite 
​ 8​
x6​ ​ ​ using radicals and using powers.
9
Using Radicals
Using Powers
__
​ 1 ​
3
3 3
6

6
5
​ ​2
​3​ ​ ?
​ ​x
​ ​ ​
5 (​23​ ​? ​x6​ ​​)​3​
5 2​x2​ ​
5 ​2​3​? ​x​3​
3


x6​ ​ ​ 5 
​ ​2​ ​? ​x​ ​ ​​
x6​ ​ ​ 5 (8​​x​6​)​3​
​ 8​

 8​
3
3
__
​ 3 ​
__
​ 1 ​
__
​ 6 ​
5 ​2​1​? ​x2​ ​
5 2​x​2​
1. Which method do you prefer?
© Carnegie Learning
My motto
is, when in doubt
rewrite radicals using
radical form!
704 Chapter 9 Radical Functions
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4 
2. Devon and Embry shared their work for extracting roots from 
​ ​f 8​ ​​g​4​ ​.
Embry
Devon
_​ 1  ​
4 
8 4
​ 
​f​ ​​g​ ​ ​
5 ​(f​   8
​ ​​g4​​)​4​
4 
8
​ ​ ​ ? 
​ ​g 4​ ​ ​
​ ​f​ ​​g​ ​ ​5 ​ ​f
4 
8  4
4
4 
5
​ (​
f  2
​ ​​)​4​ ​
?
​ ​g 4​ ​ ​
4
5 f​ 2​​|g|
_​ 1  ​
_​ 1  ​
5 (​​f  8
​ ​)​4​(g​ 4​​​)​4​
5 f​ ​4 ​​g​4 ​
5 f​   2
​ ​|g |
9
​ __ 8 ​  __
​  4 ​
a. Explain why it is not necessary to use the absolute value symbol around ​f 2​ ​.
b. Explain why it is necessary to use the absolute value
symbol around g.
When
the power and
root are equal, even
numbers, remember to use
absolute value for the
principal root.
© Carnegie Learning
4 
In Question 2, Embry extracted the root from 
​ ​f 8​ ​​g​4​ ​
using radical
form because the root of a product is equal to the product of its
p m
p n
m n
roots, 
​  p ​a
​ ​​b​ ​ ​
5
​    ​a
​ ​ ​ ? 
​    ​b
​ ​ ​.
That concept applies to quotients also. The root of a quotient
p m
m


____
is equal to the quotient of its roots, ​  p ___
​ ​a​n ​ ​​  5
​ ​ p ​a​ ​ ​ ​  .
n
​b​ ​ 
​ ​b
​ ​ ​

9.4 Extracting Roots and Rewriting Radicals 504368_A2_Ch09_661-736.indd 705
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For some radicals, you may not be able to extract the entire radicand.
3

3. Angelo, Bernadette, and Cris extracted the roots from 
​ 16​
x8​ ​ ​.
Angelo
Bernadette
__1
9
6
__1
2 ​ 3  ​
__1
__1
5 (​2​​? 2 ? x​ ​ ​? x​ ​ ​​)​​
5 (​​2​3​)​ ​3  ​ ​? ​​(2​1​)​ ​3  ​​  ? (​x​ 6​​)​​ 3  ​ ​? (​x​ 2​​)​​ 3  ​​
5 ​2​1​? ​2​ ​3  ​ ​? x​  2​ ​? x​ ​ ​3  ​​
3  2
 ? x​  2
5 2 ?
​3 2 ​
​ ​?
​ ​x
​ ​ ​
3

5 2​x 2​ ​
​ 2​
x 2​ ​ ​
3
__1
__4
__8

​3  16​
x8​​ ​ 5 ​2​ ​3  ​​? x​ ​ ​3  ​​

3

​ 16​
x 8​​ ​
5 ​(​16x 8​​)​ ​3  ​ ​

__1
5 ​(​2​4​​x8​​)​ ​3  ​ ​
__1
__1
2__
__3
__1
__6
__2
5 ​2​ ​3 ​ ​? ​2​ ​3  ​ ​? x​ ​ ​3  ​​? ​x​ ​3  ​​
5 2 ?
​  x ​ ? x​ 2​​? ​  x ​
3
3
3
5 2​x 2​ ​
​  2​
x 2​ ​ ​
Cris
​  
16​x8​ ​  5 
​  
​24​ ​? x​ 8​ ​

3
3
5
​  
​23​ ​x​ 6​ ​
?
​  
2​x 2​ ​
3
5 2​x  2
​ ​
​  
2​x 2
​ ​
3
3
b. Compare and contrast the methods.
706 © Carnegie Learning
a. In the last line of work, why was ​2x​2​not extracted from the radical?
Chapter 9 Radical Functions
504368_A2_Ch09_661-736.indd 706
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?
____
4. Betty, Wilma, and Rose each extracted roots and rewrote the radical √
​ ​x2​ ​​y​2​ ​.
Betty
___
____
​√​x2​​​y2​​ ​ 5 √
​ ​x2​​? y​ 2​​ ​
__ __
5√
​ ​x2​​ ​ ? √
​ ​y2​​ ​
___
​x2​​​y2​​ ​ 5
​√
_____
​ ​x2​​? y​ 2​​ ​
__
__
√
​ ​x2​​ ​ ? ​ ​y2​​ ​
√
5
5 |xy|
5 |x|?|y|
√
Rose
____
______
2 2
2
​ ​x​ ​​y​ ​  5 ​ __
​x2​ ​? y​ __
​ ​
5 √
​ ​x2​ ​  ? ​ ​y2​ ​
5 xy
√
√
√
9
Who’s correct? Explain you reasoning.
© Carnegie Learning
Wilma
9.4 Extracting Roots and Rewriting Radicals 504368_A2_Ch09_661-736.indd 707
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5. Rewrite each radical by extracting all possible roots, and write the final answer in
radical form.
_____
a.​√ 16​x6​ ​ ​
___
b. 2​√ 8​v3​ ​
9
____
c.​√​d3​ ​​f​4​ ​
____
d.​√ ​h4​ ​​j​ 6​ ​
​
_________
e.​√ 25​a2​ ​​b​8​​c​10​ ​
4 
x5​ ​​y​12​ ​
f.​
 81​
​
h.​√(x 1 ​3)​2​
​
​
© Carnegie Learning
_______
3
g.​
1 ​3)​9​
 (x
Be prepared to share your solutions and methods.
708 Chapter 9 Radical Functions
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Time to Operate!
9.5
Multiplying, Dividing, Adding,
and Subtracting Radicals
Learning Goals
In this lesson, you will:
• Rewrite radicals by extracting roots.
• Multiply, divide, add, and subtract radicals.
T
he word radical can describe something that is cool, something that is extreme
or very different from the usual, something related to the root or origin in a
non-mathematical context, and of course a mathematical function.
© Carnegie Learning
The origin or the word radical is related to the Latin word radix, meaning “root.”
709
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Problem 1 Multiplying and Dividing Radicals
_____
___
1. Arianna and Heidi multiplied √
​ 18​a2​ ​ ​
?4√
​ 3​a2​ ​ ​ and extracted all roots.
Arianna
9
____
___
_____
√
​ ​18a 2​ ​ ​ ? 4​√​3a 2​ ​ ​ 5 4​√​54a 4​ ​ ​
________
5 4​√9 ? 6 ? a​ 4​ ​
​
54?√
​ 9 ​ ? √
​ 6 ​ ? √
​ ​a 4​ ​ ​
54?3?√
​ 6 ​ ? a​ 2​ ​
5 12​a 2​ √
​​ 6 ​
__
__
__
__
__
Heidi
___
__
_____
____
​√18​a2​​ ​ ? 4​√3​a2​​ ​ 5 √
​ 9 ? 2 ? a​ 2​​ ​
?4?√
​ 3 ? a​ 2​​ ​
_ __2
_ _ __2
5√
​ 9 ​ ? √
​ 2 ​ ? √
​ ​a​​ ​ ? 4 ? √
​ 3 ​ ? √
​ ​a​​ ​
_
_
53?√
​ 2 ​ ? |a| ? 4 ? √
​ 3 ​ ? |a|
_
5 12​a2​ ​√
​ 6 ​
Compare Arianna’s and Heidi’s solution methods. Explain the difference in their
solution methods.
In a quotient, you can extract roots using different methods.
Exponential Form
_____
__
​ 1 ​
Radical Form
_____
________
4​√3​a​ ​ ​
4​√3​a​ ​ ​
(18​a​ ​​)​ ​
√ ___
√ 3 ? ​a​ ​ ​
a​ ​ ​
a​ ​ ​
_________
​√ 18​
______
___
​
5 ______
​
​5 ​ ​ 6 ? ___
​
​ ​ 18​
​​ ______
1
__
2 2
4​√3​a​ ​ ​  4(3​​a​2​)​ ​2 ​ ​
2 __
​ 1 ​
1  ​? ​​_____
5 ​ __
​ 18​a2​ ​ ​ ​​2​
4
3​a​ ​
1
__
5 __
​ 1 ​ ? ​6​ ​2 ​ ​
4
__
1
__
5 ​    ​? √
​ 6 ​
4__
​√ 6 ​
5 ​ ___
​
4
2
(  )
710 2
2
2
2
__
√
5 ___
​ ​ 6 ​
​
4
I wonder
if it would be
better to extract roots
from each radical first,
then divide out common
factors?
© Carnegie Learning
2
Chapter 9 Radical Functions
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2. Which method do you think is more efficient?
_____
9
√
3. Jackie shared his solution for extracting roots and rewriting the quotient ______
​ ​3 25bc ​
​
,
2 2
​ ​b
​ ​​c​ ​ ​

given b . 0 and c . 0.
Jackie
___
___
√
​​ √25bc ​ ​
​
_____
_____
5 ​   3 252bc ​
​
3 2 2
2
​  ​b
​​​c​​ ​  
​  ​b
​ ​​c​​ ​

__
√
​
25 ​
____
5 ​   3  ​
​  bc ​


5
​  bc ​


5 ____
​      ​
3
a. Why are the restrictions b . 0 and c . 0, instead of b $ 0 and c $ 0?
© Carnegie Learning
b. Explain why Jackie’s work is incorrect.
9.5 Multiplying, Dividing, Adding, and Subtracting Radicals 504368_A2_Ch09_661-736.indd 711
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?
c. Robert and Maxine
also shared their solutions for extracting roots and rewriting
_____
√
25bc ​
​
the quotient ______
​  3
​
, given b . 0 and c . 0.
2 2
​ ​b
​ ​​c​ ​ ​

9
Maxine
____
___
__
​√25bc ​  _________
√  ? √
​ bc ​
______
​ 3
​
5 ​ ​ 25 ​
​
3
2 2
​   ​b
​ ​​c​ ​ ​

2 2
​  __
​b
​ ​​c​ ​ ​

5​√bc ​   ​
5 ​ ______
3 2 2
​   ​b
​ ​​c​ ​ ​

Who’s correct? Explain your reasoning.
© Carnegie Learning
Robert
___
__ __
√
√
25
bc ​
________
? √
​  ​
bc ​
​
​​ _____
3 2 2   ​5 ​  25 ​
3 2 2
​  ​b
​​​c​​ ​  
​__
​
b
​​​c​​ ​

√
5 _____
​   5​3  2bc ​2   ​
​ ​
b
​​​c​​ ​

__
3
√   

​  bc ​
5 _____
​   5​3 bc ​
​ ? ____
​
​
3
2
2

​  bc ​
​  __
​b
​​​c​​ ​  

3
√
​   ​
bc ​

5 _________
​ 5​ 3bc ​   3
3
​  ​b
​ ​​c​ ​ ​

__1
__​ 1  ​
2​(b​c)​ ​3  ​ ​
bc​
)
​
5(
________
5 ​
bc ​
_​ 5  ​
bc​
)
​6​
5(
_____
5 ​   ​
bc
5 5 _​ 61  ​
​​​)​ ​
5 ______
​ 5(​b​​​c ​
bc
6 5 5
​​​c​​ ​
5
​  ​b

_______
​
5 ​
bc
712 Chapter 9 Radical Functions
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4. Perform each operation and extract all roots. Write your final answer in radical form.
__
__
__
a. 2​√ x ​ ? √
​ x ​ ? 5​√ x ​ , given x $ 0
9
3
(
)
b. 2
​3 k ​
​  k ​
__
__
__
c. 7​√ h ​(3​
√ h ​ 1 4​√ ​h3​ ​ ​),
given h $ 0
Hmm . . ., is it
better to extract
the roots using radical
form or exponential
form?
__
3
d.​√ a ​ ? 
​ a ​
, given a $ 0

3
© Carnegie Learning

)( 
e. (n
​3 4n ​
​ 2​
n2​ ​ ​ )
____
√
4
f.​ ___
​ 4​x2​ ​ ​​ , given x fi 0
​x​ ​
9.5 Multiplying, Dividing, Adding, and Subtracting Radicals 504368_A2_Ch09_661-736.indd 713
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5. Perform each operation, extract all roots, and write your final answer in radical form,
without
radicals in the denominator.
___
__
√ 2​a5​ ​ ​
√
a ​
2​
2​
_____
_______
​
, given a . 0
a.​  3
b.​  3
​,
given a . 0

5 ​

 a ​
5
​ 16​
a2​ ​ ​
9
Multiplying by
a form of one
helps to eliminate
the radical from
the denominator.
_____
25​√4​j​ ​​k​ ​ ​
_____ ​
, given j . 0 and k . 0
c.​ _________
2 5
© Carnegie Learning
​√75​jk​2​ ​
714 Chapter 9 Radical Functions
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Problem 2 Adding and Subtracting Radical Terms
To add and subtract terms, it is important to identify like terms.
1. Use the symbols to identify six groups of like terms. The first group has been started
for you.
9
1
4x
1 5
x
3
2x3
10√x2
x3
√x
1
26x 5
25x3
23x
√x
3
28√x
3
2
© Carnegie Learning
27√x
3
2√x
1
√x
3
3
3
10√x
√x2
0.2x
1
x5
2√x
x
9.5 Multiplying, Dividing, Adding, and Subtracting Radicals 504368_A2_Ch09_661-736.indd 715
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In some cases, you can rewrite the sum or difference of two terms as one term.
2. Explain why Grace and Diane were able to rewrite their original expression as one term.
Grace
_
_
Diane
_
2​√x ​ 1 6​√x ​ 5 8​√x ​
9
3 2
3 2
3 2
16 
​ ​
x​​ ​ 2 10 
​ ​
x​​ ​ 5 6 
​ ​
x​​ ​
Ron
3
3
0.1​√__x ​  1 3.6 
​  
x ​ 5 3.7 
​  
x ​
Sheila
_
_
__
​√x ​ 1 √
​ y ​ 5 2​√ xy ​
4. Explain why Sheila’s answer is incorrect.
716 © Carnegie Learning
3. Explain why Ron’s answer is incorrect.
Chapter 9 Radical Functions
504368_A2_Ch09_661-736.indd 716
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When adding or subtracting radicals, you can combine like terms and write the result using
fewer terms.
__
__
__
For example, the two terms, 3​√ x ​ and √
​ x ​ are like terms because their variable portions, √
​ x ​ ,
are the same. The coefficients do not have to be the same.
3
3
4
On the other hand, the terms 28
​ ​x
​ ​ ​ and 7
​ x ​
 are not like terms because their variable
3 4
3

portions, 
​ ​x​ ​ ​ and 
​ x ​
, are different. The indices are the same but the radicands are different.

__
__
​ 4 ​
​ 1 ​
In exponential form, ​x​3​and ​x​3​, notice that the bases are the same, the denominators in the
exponent are the same, but the numerators in the exponents are different.
9
To determine the sum or difference of like radicals, add or subtract
the coefficients.
__
__
__
3​√ x ​ 1 √
​ x ​ 5 4​√ x ​ , given x $ 0
You can also write an equivalent expression using powers.
1
__
1
__
1
__
3​x​​ 2 ​ ​1 ​x​ ​2 ​ ​5 4​x​ ​2 ​ ​, given x $ 0
?
__
1
__
5. Larry and D.J. discussed whether or not 4​√ x ​ and 25​x​ ​2 ​ ​are like terms, given x $ 0.
Larry
D.J.
They are not like terms because
their variable parts are different.
They are like terms. Their
variable parts look different, but
they are actually the same.
© Carnegie Learning
Who’s correct? Explain your reasoning.
9.5 Multiplying, Dividing, Adding, and Subtracting Radicals 504368_A2_Ch09_661-736.indd 717
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14/11/13 5:51 PM
6. Combine like terms, if possible, and write your final answer in radical form.
__
__
a.​√ y ​ 2 √
​ y ​ , given y $ 0
9
__
__
b. 9​√ a ​ 1 5​√ b ​ , given a $ 0, b $ 0
__
__
__
c. 2​√ x ​ + √
​ x ​ 1 5​√ x ​ , given x $ 0
__
__
__
d. 7​√ h ​ 2 4.1​√h ​ 1 2.4​√ h ​,  given h $ 0
_
_
_
__
3
f. 5​√ g ​ 1 2
​ g ​
, given g $ 0

718 © Carnegie Learning
e. 3​√ t ​ (​√ t ​ 2 8​√ t ​ ) 1 4t, given t $ 0
Chapter 9 Radical Functions
504368_A2_Ch09_661-736.indd 718
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Talk the Talk
Complete the graphic organizer. Write two radicals whose sum, difference, product, and
3
quotient are each equivalent to 6 
​ x ​
.

Difference
Sum
9
3

6
​  x
​
Quotient
© Carnegie Learning
Product
Be prepared to share your solutions and methods.
9.5 Multiplying, Dividing, Adding, and Subtracting Radicals 504368_A2_Ch09_661-736.indd 719
719
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© Carnegie Learning
9
720 Chapter 9 Radical Functions
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9.6
Look to the Horizon
Solving Radical Equations
Learning Goals
In this lesson, you will:
• Use algebra to solve radical equations.
• Write the solution steps of a radical equation using radical notation.
• Write the solution steps of a radical equation using exponential notation.
• Identify extraneous root when solving radical equations.
S
o, you have been wondering whether there is a system to measure wind speed
and describe conditions at sea and on land, right? The answer is the Beaufort
scale. It was developed in the early 1800’s and is still in use today.
© Carnegie Learning
Beaufort Scale
Beaufort Number
Description
Wind Speed
(miles per hour)
Wave Height
(feet)
0
calm
,1
0
1
light air
1–3
0–1
2
light breeze
4–7
1–2
3
gentle breeze
8–13
2–3.5
4
moderate breeze
13–17
3.5–6
5
fresh breeze
18–24
6–9
6
strong breeze
25–30
9–13
7
near gale
31–38
13–19
8
gale
39–46
18–25
9
strong gale
47–54
23–32
10
storm
55–63
29–41
11
violent storm
64–73
37–52
12
hurricane
74
46
721
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Problem 1 Analyzing Solution Paths for Radical Equations
Strategies for solving equations such as maintaining balance and isolating the term
containing the unknown are applicable when solving radical equations.
Let’s compare the algebraic solution of a two-step quadratic equation to a two-step
radical equation.
9
Solution Steps for a
Quadratic Equation
Check x 5 3:
2​x​2​2 5 5 13 2​x2​ ​5 18
2(​3)​2​2 5 0 13
​x2​ ​5 9
13 5 13 �
__
__
√
​ ​x2​ ​ ​ 5 √
​ 9 ​
Check x 5 23
x 5 63
2(​23)​2​2 5 0 13
13 5 13 �
__
Solution Steps for a
Radical Equation
2​√ x ​ 2 5 5 13
__
2​√x ​ 5 18
__
√
​ x ​ 5 9
__
(​√ x ​​ )​2​5 (9​)2​ ​
x 5 81
Check
x 5 81:
___
2​√ 81 ​ 2 5 0 13
13 5 13 �
1. Analyze the examples.
a. Describe the similarities in the first two steps of each solution.
© Carnegie Learning
b. Describe the differences in the remaining steps of each solution.
722 Chapter 9 Radical Functions
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__
2. Franco, Theresa, Dawnelle, and Marteiz shared their work for solving 3​√ x ​ 1 7 5 0,
given x $ 0.
Franco
Theresa
_
3​√x ​ 1 7 5 25
(3​√x ​ 1 7​)2​​5 ​(25)​2​
_
_
9x 1 42​√x ​ 1 49 5 625
_
9x 1 42​√x ​ 2 576 5 0
3(3x 1 14​√x ​ 2 192) 5 0
3(3​√x ​ 1 32)(​√x ​ 2 6) 5 0
_
_
_
_
_
3√
​ x ​ 1 32 5 0 or ​√x ​ 2 6 5 0
_
_
3​√x ​ 5 232​√x ​ 5 6
_
_
232
​√x ​ 5 ____
​  3 ​
(​√x ​​ )​2​5 (6​)2​​
_
232 2
(​√x ​​ )​2​5 ​​ ____
x 5 36
​  3 ​
​​​
1024
x 5 ____
​  9 ​
(  )
_
3​√x ​ 1 7 5 25
_
(3​√x ​ 1 7​)2​​5 (25​)2​​
_
9x 1 42​√x ​  1 49 5 625
_
9x 1 42​√x ​  2 576 5 0
_
3(3x 1 14​√ x ​ 2 192) 5 0
_
_
3(3​√x ​  1 32)(​√x ​  2 6) 5 0
_
_
3​√x ​ 1 32 5 0 or ​√x ​ 2 6 5 0
_
_
3​√x ​ 5 232​√x ​  5 6
_ 232
_
​√x ​ 5 ____
(​√x ​​ )2​​5 (6​)2​​
​  3 ​
2
_
​ 2332 ​  ​​​
x 5 36
(​√x ​​ )2​​5 ​​ ____
x 5 ____
​ 1024
9 ​
Check:
____
___
3​ ​( ____
​
​ 1024
​ ​
1 7 0 25 3​√(36) ​ 1 7 0 25
9
​ 32
3​ __
3 ​   ​1 7 0 25 3(6) 1 7 0 25
39 fi 25
25 5 25 ✓
There is one solution, x 5 36.
(  )
© Carnegie Learning
√
)
(  )
Dawnelle
3​√__x ​  1 7 5 25
(3​√__x ​1 7​)2​ ​5 (25​)2​ ​
3x 1 7 5 625
3x 5 618
x 5 206
Marteiz
_
3​√x ​ 1 7 5 25
_
Check:
____
3​√x ​ 5 18
3​√(36) ​ 1 7 0 25
​√x ​ 5 6
3(6) 1 7 0 25
_
_
(​√x ​​)​ 2​5 (6​)2​ ​
25 5 25 ✓
x 5 36
9.6 Solving Radical Equations 504368_A2_Ch09_661-736.indd 723
9
723
14/11/13 5:51 PM
a. Theresa and Marteiz each solved the equation correctly. Explain the difference
between their solution methods.
9
b. Explain the error in Franco’s work.
c. Explain the error in Dawnelle’s work.
3. Solve and check each equation.
___
a.​√ 2x ​ 5
3
Increasing the
power of the
variable may introduce an
extraneous solution . . .
So, remember to check
your answers.
_____
c. 4​√ x 2 6 ​
58
724 © Carnegie Learning
3

b.​
2 3 ​
52
 2x
Chapter 9 Radical Functions
504368_A2_Ch09_661-736.indd 724
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_______
d.​√2x 1 1 ​
55
e. 2
​3 x ​
 1 16 5 0
9
_______
f.​√3x 2 1 ​
1958
__
© Carnegie Learning
g. x 2 √
​ x ​ 5 2
______
h. x 2 1 5 √
​ x 1 1 ​
9.6 Solving Radical Equations 504368_A2_Ch09_661-736.indd 725
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Problem 2 Read, Interpret, and Solve
1. The Beaufort scale is a system that measures wind speed and describes conditions at
sea and on land. The scale’s range is from 0 to 12. A zero on the Beaufort scale means
that the wind speed is less than 1 mile per hour and the conditions at sea and on land
are calm. A twelve on the Beaufort scale represents hurricane conditions with wind
speeds greater than 74 miles per hour, resulting in greater than 50-foot waves at sea
and severe damage to structures and landscape.
9
3
__
Consider the equation V 5 1.837​B​ ​2 ​ ​that models the relationship between wind speed in
miles per hour V and the Beaufort numbers B. Determine the Beaufort number for a
wind speed of 20 miles per hour.
© Carnegie Learning
2. In medicine, Body Surface Area BSA is ______
used to help determine proper dosage
√
W
? ​
H ​
​
_______
for medications. The equation BSA 5 ​
models the relationship between BSA
60
in square meters, the patient’s weight W in kilograms, and the patient’s height H in
centimeters. Determine the height of a patient who weighs 90 kilograms and has a
BSA of 2.1.
726 Chapter 9 Radical Functions
504368_A2_Ch09_661-736.indd 726
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3. Big Ben is the nickname of a well-known clock tower in London, England, that stands
316 feet tall. The clock is driven by a 660-pound pendulum in the tower that continually
swings back and forth. The relationship between the length of pendulum L in feet and
the time it takes for a pendulum to___
swing back and forth one time, or its period T, is
L
modeled by the equation T 5 2p​ ___
​    ​ ​ .  If the pendulum’s period is 4 seconds, determine
32
the pendulum’s length.
√
© Carnegie Learning
9
9.6 Solving Radical Equations 504368_A2_Ch09_661-736.indd 727
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4. A pilot is flying a plane high above the earth. She has
clear vision to the horizon ahead.
a. Use the diagram to derive an equation to show the
relationship between the three sides of the triangle.
Then, solve the equation for the plane’s altitude, p.
9
Note: The variable r represents the Earth’s radius
(miles), p represents the plane’s height above the
earth, or altitude (miles), and h represents the
distance from the pilot to the horizon (miles).
Earth’s
Surface
The Earth
is not actually a
perfect sphere, but it’s
very close. Our work will give
us a very good estimate of the
distance from the pilot to
the horizon.
p
h
r
r
© Carnegie Learning
Earth’s
Center
728 Chapter 9 Radical Functions
504368_A2_Ch09_661-736.indd 728
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b. Use your equation from part (a) to calculate the plane’s altitude, if the distance from
the pilot to the horizon is 225 miles. The earth’s radius is 3959 miles.
© Carnegie Learning
9
Be prepared to share your solutions and methods.
9.6 Solving Radical Equations 504368_A2_Ch09_661-736.indd 729
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© Carnegie Learning
9
730 Chapter 9 Radical Functions
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Chapter 9 Summary
Key Terms
• inverse of a function (9.1)
• invertible function (9.1)
• Horizontal Line Test (9.1)
9.1
• square root function (9.2)
• cube root function (9.2)
• radical function (9.2)
• composition of functions (9.2)
9
Graphing Inverses of Power Functions
A function f is the set of all ordered pairs (x, y) or (x, f(x)), where for every value of x, there is one
and only one value of y, or f(x). The inverse of a function is the set of all ordered pairs (y, x), or
(f(x), x). To graph the inverse of a function, simply reflect the function over the line y 5 x.
Example
Graph f(x) 5 ​x3​ ​, and then graph its inverse.
y
f(x) 5 x3
y5x
f21(x)
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x
731
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Determining Whether or Not Functions are Invertible
9.1
To determine whether or not a function is invertible, graph the function and apply the
Horizontal Line Test. If the graph of the function passes the Horizontal Line Test, then it is
invertible.
Example
4
Determine whether or not f(x) 5 ___
​ ​x​ ​  ​ is invertible.
56
9
y
x
4
The function f(x) 5 ___
​ ​x​ ​  ​ is not invertible, because it fails the Horizontal Line Test. That is, a
56
horizontal line can pass through more than one point on the graph at the same time.
Determining the Equation for the Inverse of a Power Function
9.2
To determine the equation for the inverse of a power function, transpose the x and the y in
the equation and then solve for y.
Determine the equation for the inverse of the function y 5 __
​ 2 ​ ​x5​ ​.
3
2
__
5
y 5 ​    ​​x​ ​
3
__
x 5 ​ 2 ​ ​y5​ ​
3
3
__
​   ​ x 5 ​y5​ ​
2

5 __
​ ​ 3 ​ x ​ 5 y
2
5

48x ​
​​ _____

5 y
​
2
5

2  ​​x5​ ​is y 5 
_____
​ ​ 48x ​
Therefore, the equation for the inverse of the function y 5 ​ __
​.
3
2

732 © Carnegie Learning
Example
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9.2
Describing the Characteristics of Square Root and Cube Root
Functions
The characteristics of square root and cube root functions include the domain, range, and
x- and y-intercepts.
Example
______
Describe the characteristics of the function f(x) 5 √
​ 5 2 x ​
9
Domain: (2`, 5]
Range: [0, `)
x-intercept: (5, 0)
__
y-intercept: (0, ​√ 5 ​ )
9.3
Describing Transformations of Radical Functions
Transformations performed on a function f(x) to form a new function g(x) can be described by
the transformational function:
g(x) 5 Af(B(x 2 C)) 1 D
Translating a Radical Function Horizontally: If a number, C, is added under the radical, the
graph of the function is shifted C units to the left. If a number, C, is subtracted under the
radical, the graph of the function is shifted C units to the right.
Translating a Radical Function Vertically: If a number, D, is added outside the radical, the
graph of the function is shifted D units up. If a number, D, is subtracted outside the radical,
the graph of the function is shifted D units down.
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Vertically Stretching and Compressing a Radical Function: Multiplying the function by a
number, A, that is greater than one vertically stretches the function. Multiplying the
function by a number, A, that is greater than zero but less than one vertically compresses
the function.
Reflecting a Radical Function: Multiplying the function by a negative one reflects the graph
across the x-axis. Multiplying by a negative one under the radical reflects the graph across
the y-axis.
Example
3
Describe how the graph of the function f(x) 5 
​ x ​
 would be transformed to produce the
graph of the function g(x) 5 2f(x 2 4) 1 1.
The graph of f(x) would be vertically stretched by a factor of two, translated 4 units to the
right and up 1 unit.
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9.3
Graphing Transformations of Radical Functions
Transformations that take place inside the radical shift the function left or right.
Transformations that take place outside the radical shift the function up or down.
Example
__
The graph of f(x) 5 √
​ x ​ is shown. Graph the transformation of f(x) as represented by the
equation g(x) 5 f(x 1 5) 1 3. Then, list the domain for each function.
9
y
8
g(x) 5 f(x 1 5)1 3
6
f(x) 5 √x
4
2
0
28 26 24 22
2
4
6
8
x
22
24
26
28
Domain of f(x): [0, `)
Domain of g(x): [25, `)
9.4
Rewriting Radical Expressions
To rewrite a radical expression, extract the roots by using the rational exponents and the
n n
properties of powers. To extract a variable from a radical, the expression 
​ ​x
​ ​ ​ can we written
as |x| when n is even, and x when n is odd.
Example
4 
Rewrite the expression 
​ 625​x6​ ​​y​5​z ​.


​ 625​
x8​ ​​y​5​z ​
5
​ 625 ? ​
x8​ ​​ ? y​4​ ? y ? z ​

734 4
4 8
4
4 
4
4

5
​ 625 ​
?
​ ​x
​ ​ ​ ? 
​ ​y4​ ​ ​ ? 
​ y ​
​ z ​
 ? 

4
5 5​x2​ ​|y|
​ yz ​

© Carnegie Learning
4
Chapter 9 Radical Functions
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9.5
Operating with Radicals
To operate on radicals, follow the order of operations and properties of powers. Remember
to extract all roots.
Example
Perform the indicated operations and extract all roots for x $ 0 and y $ 0. Write your final
answer in radical form.
____
__
(
) (
)
1 ​
​ __
__
​ 1 ​
9
__
​ 1  ​
4 
(23x​√ ​x5​ ​​y​6​ ​)(2​
√ ​x3​ ​ ​)  2 5
​ 81​x20
​ ​​y​12​ ​
5 ​23x(​x5​ ​​y​6​​)​2​ ​​2(​x3​ ​​)​2​ ​2 5(81​x20
​ ​​y​12​​)​4​
9.6
5
__
6
__
3
__
20
___
1
__
12
___
5 (23 ? x ? ​x​​ 2 ​ ​? ​y​ ​2 ​ ​)(2 ? ​x​ ​2 ​ ​) 2 (5 ? ​81​​ 4 ​ ​? ​x​ ​4 ​ ​? ​y​ ​4 ​ ​)
5 (23 ? ​x​​ 2 ​ ​?​y3​ ​)(2 ? ​x​ ​2 ​ ​) 2 (5 ? 3 ? ​x5​ ​? ​y3​ ​)
5 (23 ? ​x​ ​2 ​ ​?​y3​ ​)(2 ? ​x​ ​2 ​ ​) 2 (15 ? ​x5​ ​? ​y3​ ​)
5 26 ? ​x5​ ​? ​y3​ ​2 15 ? ​x5​ ​? ​y3​ ​
5 26​x5​ ​​y​3​2 15​x5​ ​​y​3​
5 221​x​5​​y​3​
7
__
3
__
7
__
3
__
Solving Radical Equations
To solve a radical equation, isolate the radical term if possible. Then, raise the entire
equation to the power that will eliminate the radical. Finally, follow the steps necessary to
solve the equation. Check for extraneous solutions.
Example
______
​√ x 1 2 ​
1 10 5 x
______
√
​ x 1 2 ​
5 x 2 10
______
(​√ x 1 2 ​​
)​2​5 (x 2 10​)2​ ​
x 1 2 5 ​x2​ ​2 20x 1 100
0 5 x​ 2​ ​2 21x 1 98
Check:
Check:
_______
√
1 10
​ 14 1 2 ​
___
​√16 ​ 1 10
______
0 14
​√ 7 1 2 ​
1 10 0 14
0 14
​√9 ​ 1 10 0 14
14 5 14 �
__
13 fi 14
Extraneous solution
© Carnegie Learning
0 5 (x 2 14)(x 2 7)
x 5 14 or x 5 7
There is one solution, x 5 14.
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9.6
Problem Solving with Radical Equations
To solve a problem with radical equations, identify what the problem is asking. Then,
determine how to use the given equation to solve the problem. Finally, follow the process for
solving radical equations.
Example
The distance between
any two points on a coordinate plane can be calculated by using the
___________________
equation d 5 √
​ (x2 2 x1​)2​ ​1 (y2
2 y1​)2​ ​ ​, where (x1, y1) represents the coordinates of one point
and (x2, y2) represents the coordinates of the other point. Determine the point(s) on the line
y 5 1 that is (are) exactly 5 units from the point (1, 22). Use the point (x, 1) to represent a
point on the line y 5 1.
9
___________________
d5√
​ (x2 2 x1​)2​ ​1 (y2
2 y1​)2​ ​ ​
__________________
5 5 ​√ (1 2 x​)2​ ​1 (22
2 1​)2​ ​ ​
________________
55√
​ (1 2 2x 1 x​ 2​
​1 9) ​
25 5 1 2 2x 1 x​ 2​ ​1 9
0 5 x​ 2​ ​2 2x 2 15
0 5 (x 1 3)(x 2 5)
x 5 23 or x 5 5
© Carnegie Learning
The points (23, 1) and (5, 1) are on the line y 5 1 and are exactly 5 units from the
point (1, 22).
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