LESSON
11.1
Name
Understanding
Rational Exponents
and Radicals
Class
Date
11.1 Understanding Rational
Exponents and Radicals
Essential Question: How are radicals and rational exponents related?
A1.11.A…simplify numerical radical expressions involving square roots. Also A1.11.B
Explore 1
Texas Math Standards
Understanding Integer Exponents
2
Recall that powers like 3 are evaluated by repeating the base (3) as a factor a number of times equal to the exponent
(2). So 3 2 = 3 ∙ 3 = 9. What about a negative exponent, or an exponent of 0? You cannot write a product with a
negative number of factors, but a pattern emerges if you start from a positive exponent and divide repeatedly by the
base.
The student is expected to:
A1.11.A
Simplify numerical radical expressions involving square roots.
Also A1.11.B

Starting with powers of 3:
3 3 = 27
Mathematical Processes
32 = 9
A1.1.F
31 = 3
Analyze mathematical relationships to connect and communicate
mathematical ideas.

1
Dividing a power of 3 by 3 is equivalent to reducing the exponent by .
Language Objective

Complete the pattern:
2.C.3, 2.C.4, 2.I.3, 2.I.4
Explain how radicals and rational exponents are related.
Essential Question: How are radicals
and rational exponents related?
Radicals and rational exponents can be converted
back and forth into one another, showing that they
are two different forms of notation for the same
mathematical idea.
PREVIEW: LESSON
PERFORMANCE TASK
View the Engage section online. Discuss the photo,
and the fact that carbon-14, a radioactive isotope of
carbon, occurs in trace amounts, making up about
1 part per trillion of the carbon in the atmosphere.
Then preview the Lesson Performance Task.

© Houghton Mifflin Harcourt Publishing Company
ENGAGE
→3 2
33 _
÷3
_ →3 1
_ →3 0
→9
27 _
÷3
_ →3
_→ 1
÷3
÷3
÷3
_ →3 -1 _ →3 -2
÷3
÷3
÷3
1
_ → __
1
_ → __
3
÷3
÷3
Integer exponents less than 1 can be summarized as follows:
Words
Numbers
3 =1
0
Any non-zero number raised to the power of 0 is 1; 0 is undefined
0
(2.4) = 1
0
Any non-zero number raised to a negative power is equal to 1 divided
by the same number raised to the opposite, positive power.
1 =_
1
3 -2 = _
2
9
3
Variables
x 0 = 1 for x ≠ 0
1 for x ≠ 0,
x -n = _
xn
and integer n.
Reflect
1.
Discussion Why does there need to be an exception in the second for the case of x = 0?
For a negative exponent, using x = 0 would put a 0 in the denominator and division by
zero is not defined.
Module 11
ges
EDIT--Chan
DO NOT Key=TX-B
Correction
Lesson 1
515
gh "File info"
made throu
must be
Date
Class
Rational
rstanding
11.1 Undenents and Radicals
Expo
Name
A1_MTXESE353879_U5M11L1.indd 515
HARDCOVER PAGES 389394
Resource
Locker
d?
ents relate
Also A1.11.B
rational expon
square roots.
radicals and
involving
How are
expressions
Question:
ical radical
Essential
lify numer
A1.11.A…simp
ent
nents
to the expon
ger Expo
of times equal ct with a
nding Inte
a number
a produ
by the
(3) as a factor
cannot write
1 Understa
repeatedly
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Explore
and divide
repeat
ted by
or an
2
e exponent
3 are evalua negative exponent,
from a positiv
a
powers like
you start
What about
Recall that
emerges if
2
3 ∙ 3 = 9.
, but a pattern
(2). So 3 =
r of factors
negative numbe
base.
Turn to these pages to
find this lesson in the
hardcover student
edition.
3:
powers of
Starting with

33 = 27
32 = 9
31 = 3
ng a power
of 3 by 3
is equivalent
1
to
ent by
reducing the expon
0
_ →3
÷3
-1
_ →3
÷3
÷3
÷3
y
g Compan
Publishin
Harcour t
n Mifflin
© Houghto
1
__
_→ 9
÷3
_→
÷3
33
_→ 1
÷3
_ →3
÷3
_ →9
÷3
27
1
1 _
1 , 3-2 = _ = 3 2
_
9
3-1 = 3
follows:
arized as
1 can be summ
ents less than
Integer expon
Words
0
is undefined
of 0 is 1; 0
the power
raised to
ro number
1 divided
Any non-ze
equal to
power is
a negative positive power.
to
raised
ite,
ro number
the oppos
Any non-ze number raised to
by the same
2
1.
-2
_ →3
1
__
_→ 3
÷3

Numbers
300 = 1
(2.4) = 1
1
1 =_
_
2
3- = 32 9
Variables
x0 = 1 for
x≠0
1 for x ≠ 0,
_
x-n = xn
r n.
and intege
x = 0?
the case of
on by
second for
tor and divisi
ion in the
denomina
to be an except put a 0 in the
d
there need
x = 0 woul
Why does
nent, using
Discussion
tive expo
For a nega
defined.
zero is not
Reflect
1.
Lesson 1
515
Module 11
L1.indd
9_U5M11
SE35387
A1_MTXE
Lesson 11.1
9
1 =_
1
1 , 3 -2 = _
3 -1 = _
9
32
3
 Dividi
n:
lete the patter
 Comp
_ →3
3
515
Resource
Locker
515
11/17/14
3:58 PM
11/17/14 3:58 PM
Explore 2
Exploring Rational Exponents
EXPLORE 1
A radical expression is an expression that contains the radical symbol, ―.
n ―
For √
a , n is called the index and a is called the radicand. n must be an integer greater than 1. a can be any real
number when n is odd, but must be non-negative when n is even. When n = 2, the radical is a square root and the
index 2 is usually not shown.
Understanding Integer Exponents
You can write a radical expression as a power. First, note what happens when you raise a power to a power.
(2 3) = (2 ∙ 2 ∙ 2) 2 = (2 ∙ 2 ∙ 2)(2 ∙ 2 ∙ 2) = 2 6, so (2 3)2 = 2 3 ∙ 2.
2
In fact, for all real numbers a and all rational numbers m and n, (a m) = a m • n. This is called the Power of a Power
Property.
n ―
A radical expression can be written as an exponential expression: √
a = a k. Find the value for k when n = 2.
n
 Start with the equation. √―a = a
( √―
Square both sides.
a ) = (a )
INTEGRATE TECHNOLOGY
Students have the option of completing the activity
either in the book or online.
k
2


Definition of square root
Power of a power property
k
2
a = (ak)2
a1 = a


2k
Equate exponents.
1 = 2k
Solve for k.
1
k= _
2
QUESTIONING STRATEGIES
What pattern can you use to evaluate negative
exponents? As the value of the exponent
decreases by 1, the value of the power is divided by
the base.
Reflect
2.
What do you think will be the rule for other values of the radical index n?
_1
Other radicals can be written as a n .
Explain 1
How can you evaluate a number written with a
negative exponent? A number with a
negative exponent can be written as the reciprocal
of the number written with a positive exponent.
Simplifying Numerical Expressions with nth Roots
For any integer n > 1 , the nth root of a is a number that, when multiplied by itself n times, is equal to a.
n ―
x= √
a ⇒ xn = a
1
The nth root can be written as a radical with an index of n, or as a power with an exponent of _
n.
An exponent in the form of a fraction is a rational exponent.
1
_
n ―

a = an
The expressions are interchangeable, and to evaluate the nth root, it is necessary to find the number, x, that satisfies
the equation x n = a.

Find the root and simplify the expression.
1
_
64 3
1
_
3 ―
64 3 = √64
3 ―

= 43
Convert to radical.
Rewrite radicand as a power.
=4
Definition of nth root

1
_
1
_
81 4 + 9 2
Convert to radicals.
1
_
1
_
――
81 4 + 9 2 =
4
Rewrite radicands as powers.
Apply definition of nth root.
Simplify.
――
√ 81 + √ 9
――
――
=√ 3 +√ 3
4
4
EXPLORE 2
© Houghton Mifflin Harcourt Publishing Company
Example 1
2
Exploring Rational Exponents
QUESTIONING STRATEGIES
When you convert between radical form and
rational exponent form, what are the
restrictions on the radicand and index? Conversions
are done for all real numbers for which the radical is
defined. The index must be a positive integer and
the power of the radicand must be an integer.
= 3 + 3
= 6
Module 11
516
Lesson 1
PROFESSIONAL DEVELOPMENT
Learning Progressions
A1_MTXESE353879_U5M11L1.indd 516
In this lesson, students extend their knowledge of exponents to the properties of
integer and rational exponents while allowing for a notation for radicals in terms
of rational exponents. Some key understandings for students are as follows:
1
n ―
__
• The definition b n = √b , where b > 1 and n is a positive integer, is used to
simplify expressions with rational exponents.
• The square root and cube root of a number can be written with rational
exponents.
11/17/14 3:58 PM
EXPLAIN 1
Simplifying Numerical Expressions
with nth Roots
QUESTIONING STRATEGIES
―
When you write ( √25 ) with a fractional
exponent, what is the denominator of the
fractional exponent? Why? 2; the square root
1.
indicates a power of __
2
3
Understanding Rational Exponents and Radicals
516
Your Turn
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Modeling
3.
Review powers and roots by reviewing
3 ――
53 = 5·5·5 = 125 and √125 = 5 with students.
Have students practice writing several similar
1
__
examples. Then present the definition of b n and
discuss the Example. Show students two special
1
1
__
__
cases: 1 n = 1 and 0 n = 0 for all natural-number
values of n.
1
_
83
_1 3 ―
8 3 = √8
3 ―
= √2 3
=4+3
=7
Simplifying Numerical Expressions
with Rational Exponents
Explain 2
_1
m
_
n ―
Given that for an integer n greater than 1, √b = b n , you can use the Power of a Power Property to define b n for any
positive integer m.
m
_
m
_
1
_
= (b n )
1
_
AVOID COMMON ERRORS
bn = b
m
Example 2
Simplify expressions with fractional exponents.
2
_
27 3
m
_
EXPLAIN 2
Rewrite radicand as a power.
2
_
3 ― 2
27 3 = ( √27 )
3 ― 2
= ( √3 3 )
= 32
Definition of cube root
© Houghton Mifflin Harcourt Publishing Company
When you simplify the rational exponent,
what does it mean if the simplified form is an
integer? If it is a fraction? If the exponent is an
integer, the final form will not contain a radical sign.
If the exponent is a fraction, the final form will
contain a radical sign.
n ―
= √b m
1
__
Definition of b n
Definition of b n
1
__
1
_
m
The definition of a number raised to the power of __
n is the nth root of the number raised to the mth power. The
power of m and the nth root can be evaluated in either order to obtain the same answer, although it is generally easier
to find the nth root first when working without a calculator.

How do you simplify 81 4 ? Determine the 4th
root of 81, or 3.
1
m_
n
= (b m) n
Power of a Power Property
m
n ―
= ( √b )
With fractional exponents with a numerator
other than 1, students may confuse the index
exponent
______
with the power. Write base index on the board for
students to use as a reference.
QUESTIONING STRATEGY
_1
_1
―
3 ―
16 2 + 27 3 = √16 + √27
―
3 ―
= √4 2 + √3 3
=2
b n = b n m
Simplifying Numerical Expressions
with Rational Exponents
1
_
1
_
16 2 + 27 3
4.
=9

3
_
25 2
Definition of b n
m
__
―
25 2 = ( √25 )
3
_
3
―――
(√ 5 )
3
2
Rewrite radicand as a power.
=
Definition of square root
= 53
=
Module 11
125
517
Lesson 1
COLLABORATIVE LEARNING
A1_MTXESE353879_U5M11L1.indd 517
Peer-to-Peer Activity
Have students work in pairs. Students take turns rolling both a red (r) and a blue
(b) number cube. After each roll, the student uses the numbers shown on the
r
__
cubes to complete the expression 64 b . Then the student simplifies the expression
or states that it cannot be simplified. The other student checks the answer and
then rolls the number cubes to decide the next expression.
517
Lesson 11.1
11/13/14 12:19 PM
YourTurn
5.
3
_
32 5
_3
5 ― 3
32 5 = ( √32 )
5 ― 3
= ( √2 5 )
6.
5
_
3
_
42 - 42
INTEGRATE TECHNOLOGY
―
―
―
―
= ( √2 ) - ( √2 )
4 2 - 4 2 = ( √4 ) - ( √4 )
_5
3
_
5
2
= 23
5
Encourage the use of graphing calculators to
check the results of simplifying numerical
radical expressions and numerical expressions with
rational exponents. Ask students to use the following
sample problems to practice entering expressions
correctly into their calculators:
3
__
3 ―
√
Enter
62 as 6^(2/3); enter 32 2 as 32^(3/2); enter
1
__
25 2 as 25^(-1/2). Make sure students understand
the importance of including parentheses due to the
order of operations.
3
2
3
= 25 - 23
= 32 - 8
=8
= 24
Elaborate
7.
Why can you evaluate an odd root for any radicand, but even roots require non-negative radicands?
Multiplying a number by a negative number changes the sign, so that in a product with
multiple factors, an odd number of negative factors results in a negative product, while
an even number of negative factors results in a positive product. Positive factors do not
change the sign of a product. There is no way to make a negative product with an even
number of identical factors. For odd roots, a negative number simply has a negative root
ELABORATE
since an odd number of negative factors results in a negative product, while a positive
number has a positive root.
8.
QUESTIONING STRATEGIES
In evaluating powers with rational exponents with values like __23 , why is it usually better to find the root
before the power? Would it change the answer to switch the order?
The nth root is a smaller number than the base, while evaluating the power of m first
When simplifying a fractional exponent with
m , will you get a different answer if
the form __
n
you find the root first and then raise the answer to
the power, or raise to a power first and then take the
root? Explain. No, the order doesn’t matter. You get
the same answer either way, although it is often
easier to take the root first.
requires finding the nth root of a larger number than the base. Roots of large numbers
can be found by guessing, but smaller numbers are more familiar (you are more likely
to simply recognize the root or pick it on the first guess) and even if a few guesses are
© Houghton Mifflin Harcourt Publishing Company
required, it is easier to check with small numbers.
No, switching the order would not change the answer.
9.
Essential Question Check-In How can radicals and rational exponents be used to simplify expressions
involving one or the other?
Radical expressions are interchangeable with exponents of the form __n1 . Powers with
rational exponents can be evaluated by converting them into radical expressions with
index n. Radical expressions with powers can sometimes be simplified by switching to
rational exponents and using the properties of powers.
Module 11
518
SUMMARIZE THE LESSON
How do you simplify an equation with a
rational exponent? If the exponent has the
1 , find the nth root of the base. If the
form __
n
m , find the nth root of the
exponent has the form ___
n
base raised to the mth power.
Lesson 1
DIFFERENTIATED INSTRUCTION
A1_MTXESE353879_U5M11L1.indd 518
2/14/15 11:41 AM
Kinesthetic Experience
As students work on a problem, suggest that kinesthetic learners write the
base, index, and power on separate small pieces of paper. Have students arrange
the pieces of paper to form the original expression. Then have students draw a
radical on a sheet of paper and move pieces of paper into their correct positions in
the radical.
Understanding Rational Exponents and Radicals
518
Evaluate: Homework and Practice
EVALUATE
Evaluate the expressions.
1. 10 -2
10 -2 = 1 2
10
1
=
100
2.
_
_
3.
Practice
Explore 1
Understanding Integer Exponents
Exercises 1–6, 20
56
-1
• Online Homework
• Hints and Help
• Extra Practice
1
=_
56
1
=_
1
56
4.
_
_
2 -4 = 14
2
1
=
16
ASSIGNMENT GUIDE
Concepts and Skills
2 -4
56 -1
(_13 )
(_13 )
-2
-2
1
_
1
_
( 3)
1
=_
1
_
()
=
2
9
Explore 2
Exploring Rational Exponents
Exercises 26–27
Example 1
Simplifying Numerical Expressions
with nth Roots
Exercises 7–10,
28
Example 2
Simplifying Numerical Expressions
with Rational Exponents
Exercises 11–19,
21–25, 29
5.
(-2) °
6.
(-2 °) = 1
3 ∙ 6 -2
=9
(_61 )
1
= 3 . (_)
36
3 · 6 -2 = 3 .
2
3
_
36
1
=_
=
12
Find the root(s) and simplify the expression.
1
_
7. 81 2
_1
8.
―
―
= √9
_1
3 ――
125 3 = √125
3 ―
= √5 3
81 2 = √81
2
© Houghton Mifflin Harcourt Publishing Company
1
_
125 3
=5
=9
9.
1
_
1
_
49 2 - 4 2
_1
_1
― ―
― ―
= √7 - √2
_1
_1
5 ―
4 ―
16 4 + 32 5 = √
16 + √32
5 ―
4 ―
= √2 4 + √2 5
49 2 - 4 2 = √49 - √4
2
1
_
1
_
10. 16 4 + 32 5
2
=7-2
=2+2
=5
=4
Module 11
519
Lesson 1
LANGUAGE SUPPORT
A1_MTXESE353879_U5M11L1.indd 519
Connect Vocabulary
―
11/13/14 12:19 PM
―
3 ―
Write the terms √5 and √n5 on the board. Point out that the first expression, √5
, is known as the square root or the second root of 5.
3 ―
Explain that, in more complicated radical expressions, such as √n5 , the “inside”
expression, n 5, is called the radicand, while the “root” (3, in the upper left of the
3 ―
radical symbol) is called the index. √n5 means the third root, or cube root, of n to
the fifth power.
519
Lesson 11.1
Simplify the expressions with rational exponents.
3
_
11. 49 2
―
―
= ( √7 )
49 2 = ( √49 )
_3
2
3 ―
8 3 = ( √8 )
5
3 ―
= ( √2 3 )
_5
3
3
3
_
3
_
13. 27 3 + 4 2
―
―
_
-1
2
=
2
= 34 + 23
= 53 + 43
= 81 + 8
= 125 + 64
= 89
= 189
16. 8
1
_
_
1
-_
3
8
_
-1
3
1
2
25
3
_
1
= _1
83
1
= 3 ―
√8
1
= 3 ―
√2 3
1
=
2
_
1
_
√―
25
1
=_
√―
5
1
_
=
_
2
_
5
2
_
2
-_
3
18. 8 3 + 8
_
2
3
8 +8
=1
_
-2
3
© Houghton Mifflin Harcourt Publishing Company
2
-_
3
17. 1
2
-_
3
3
2
=
1
―
―
―
―
= ( √5 ) + ( √4 )
_3
_3
3
3
25 2 + 16 2 = ( √25 ) + ( √16 )
Simplify the expressions.
25
3
_
14. 25 2 + 16 2
_3
_4
3
3 ― 4
27 3 + 4 2 = ( √27 ) + ( √4 )
3
3 ― 4
= ( √3 3 ) + ( √2 2 )
1
-_
2
Circulate as students solve the problems. Invite
students to explain their reasoning as they begin a
new problem.
= 32
= 343
15. 25
5
= 25
= 73
4
_
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Communication
5
_
12. 8 3
_
_2
1
= 8 3 + _2
83
3 ― 2
= ( √8 ) +
1
_
―
( √8 )
3 ―
= ( √2 3 ) +
2
1
=2 +_
2
1
=4+_
4
1
_
=4
2
3
2
1
_
―
( √2 )
3
3
2
2
4
Module 11
Exercise
A1_MTXESE353879_U5M11L1.indd 520
Lesson 1
520
Depth of Knowledge (D.O.K.)
Mathematical Processes
1–23
2 Skills/Concepts
1.E Create and use representations
24–25
2 Skills/Concepts
1.A Everyday life
26
2 Skills/Concepts
1.G Explain and justify arguments
27–28
3 Strategic Thinking
1.G Explain and justify arguments
29
3 Strategic Thinking
1.F Analyze relationships
20/02/14 7:58 AM
Understanding Rational Exponents and Radicals
520
1
_
25 2
19. _
1
_
27 3
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Math Connections
20. 7 · 10 ⁻3
1
_
10
1
_
= 7 . 1000
7
=_
_
√―
25
25
_
_
_= ―
1
2
27
Explain that, when writing an expression with a
rational exponent as a radical, the power can also
be placed under the radical sign.
√27
1
3
3
―2
√5
_
―
√3
5
2
= _ = 1_
=
3
3
3
2
__
21.
For example, 216 3 can be written as
3 ――
3 ―― 2
(√
216 ) or √2162 . However, it is usually more
convenient to evaluate the root and then evaluate
the power.
7 · 10 ⁻3 = 7 .
1000
3
(_14 )
_
1
(_14 ) = _
1 _
_
( 4 )1
_
=
( √―_41 )
1
=_
1
_
( √―4 )
1
=_
1
_
( √―2 )
1
=_
1
_
()
3
⁻_
2
⁻
3
2
3
2
22. 2 · 36
1
⁻_
2
2 · 36
_
+ 6 ⁻1 =
1
2
_
_
_+
=
2
_
_1
―+
1
36 2
√36
61
6
2
_1
=_
― +
√6 2
1
2
= +
6
6
3
Students will sometimes multiply the base by the
negative exponent. Have these students re-read the
definition of a negative exponent. Point out that
10-3 means a number less than one, not a number
less than zero.
+ 6 ⁻1
⁻1
2
3
AVOID COMMON ERRORS
3
_ _
=
3
6
_1
2
2
3
2
=
1
_
1
_
()
8
© Houghton Mifflin Harcourt Publishing Company
=8
_3
23. Geometry The volume of a cube is related to the area of a face by the formula V = A 2 .
What is the volume of a cube whose face has an area of 100 cm 2?
_3
v = 100 2
――
――
= ( √10 )
= ( √100 )
2
= 1000 cm 3
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Lesson 11.1
3
= 10 3
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3
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Lesson 1
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24. Biology The approximate number of Calories, C, that
QUESTIONING STRATEGIES
3
_
an animal needs each day is given by C = 72m 4 , where
m is the animal’s mass in kilograms. Find the number
of Calories that a 16 kilogram dog needs each day.
What generalization can you make when the
radicand’s exponent and the index are equal,
4 ―
as in √34 ? When the exponent in the radicand and
the index are the same, the expression simplifies to
4 ―
the base of the radicand. Thus, √34 simplifies to 3.
_3
C = 72(16) 4
4 ― 3
= 72( √16 )
4 ―
= 72 √2 4
(
)
3
= 72 . 2 3
= 72 . 8
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Modeling
= 576 Calories
25. Rocket Science Escape velocity is a measure of how fast an object must be moving
to escape the gravitational pull of a planet or moon with no further thrust. The escape
velocity for the moon is given approximately by the equation
d ⁻_21 , where v is the escape velocity in miles per hour and d is the
V = 5600 ∙ _
1000
distance from the center of the moon (in miles). If a lunar lander thrusts upwards until it
reaches a distance of 16,000 miles from the center of the moon, about how fast must it be
going to escape the moon’s gravity?
(
v ≈ 5600 ∙
Make sure students understand that the denominator
in the exponent determines the index in the radical
1
__
equivalent. For example, in 16 4 , the 4 in the
exponent indicates the 4th root of 16.
)
16,000
(_
1000 )
_
⁻1
2
1
= 5600 ∙ ___
_1
16 2
1
_
= 5600 ∙ ___
√
16
1
= 5600 ∙ _
= 1400
4
It needs to be moving at about 1400 miles per hour.
26. Multiple Response Which of the following expressions cannot be evaluated? b, e and f are undefined.
1
_
a. 4 2
b. (-4)
c.
1
⁻_
2
4 ⁻2
⁻2
d. (-4)
e. 0
f.
© Houghton Mifflin Harcourt Publishing Company • ©WilleeCole/Alamy
= 5600 ∙ 16
1
⁻_
2
1
⁻_
2
0 ⁻2
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Lesson 1
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Understanding Rational Exponents and Radicals
522
COLLABORATIVE LEARNING
H.O.T. Focus on Higher Order Thinking
2
_
27. Explain the Error Yuan is asked to evaluate the expression (-8) 3 on his exam,
and writes that you cannot evaluate a negative number to an even fractional power.
Is he correct, and if so, why? If he is not correct, what is the correct answer?
No, he is not correct. It is only even numbers in the denominator of
an exponent that cannot be evaluated with a negative base. With an
odd denominator and an even numerator in the exponent, there is no
problem. The correct answer is:
Have students work in groups of three or four. To
help students master the concepts in the lesson, have
members of a small group create problems involving
rational exponents or radicals, then have them solve
each other’s problems. Encourage students to make
their problems as elaborate as they like, but they must
be able to supply the correct answer for any problem
that they submit.
――
_2
3
(-8) 3 = ( √
-8 )
=
2
(-2) )
( √――
3
3
2
= (-2)
2
=4
28. Communicate Mathematical Ideas Show that the nth root of a number, a, can be
1
expressed with an exponent of _
n for any positive integer, n.
n ―
√
a = ak
JOURNAL
In their journals, have students write the steps they
5
__
would use to simplify 729 6 .
Raise both sides to the nth power.
Definition of nth root a = (a k)
n
Power of a Power Property
Equate Exponents.
1
k=_
Solve for k.
n ―
a ) = (a k)
(√
n
n
a 1 = a kn
1 = kn
n
29. Explain the Error Gretchen thinks she has figured out how to evaluate the square
root of a negative number. Explain why her solution is flawed.
1
_
© Houghton Mifflin Harcourt Publishing Company
( -1 )2 ( -1 ) 2
= ( -1 )
= ( -1 )
=1
1
2·_
2
0
_
1
_
Then she solves for ( -1 ) 2 which is the same thing as √-1 .
( -1 )2
1
_
∙ ( -1 ) 2 = 1
( -1
1
_
)2
1
=_
2
( -1 )
1
=_
1
=1
But the square root of -1 cannot be 1, since 1 ∙ 1 = 1, not -1.
Gretchen's method has no validity because it is based on calculations
involving the square root of a negative number, which is not defined. It is
also riddled with errors.
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Lesson 11.1
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Lesson 1
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Lesson Performance Task
CONNECT VOCABULARY
Some students may not be familiar with the term
half-life t __1 used in this Lesson Performance Task.
( )
Carbon-14 dating is used to determine the age of
archeological artifacts of biological (plant or animal)
origin. Items that are dated using carbon-14 include
objects made from bone, wood, or plant fibers. This
method works by measuring the fraction of carbon-14
remaining in an object. The fraction of the original
carbon-14 remaining can be expressed by the function,
f = 2(
died.
t
⁻_
5700
2
Explain that the half-life of any substance is the time
it takes for the amount of the substance to decrease to
half of what it originally was. To illustrate, draw a
long line on the board, then draw successively shorter
1 as long;
lines: half as long as the original line; then __
4
1 as long; and so on. The lines represent the
then __
8
shrinking by half at equal time intervals.
), where t is the length of time since the organism
a. Fill in the following table to see what fraction of the original
carbon-14 still remains after the passage of time.
t
t
_
5700
0
0
5700
1
11,400
2
17,100
3
Fraction of Carbon-14
Remaining
1
_1
2
_1
4
_1
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Modeling
8
c. Write the corresponding expression for the remaining fraction
of uranium-234, which has a half-life of about 80,000 years.
a. f = 2
5700
(⁻_
5700 )
,
= 2 ⁻1
=
f=2
_1
2
11,400
(⁻_
5700 )
,
= 2 ⁻2
_1
2
1
=_
=
2
4
f=2
17,100
(⁻_
5700 )
= 2 ⁻3
1
_
2
1
=_
=
3
8
_
_
1
1
b. From the table, f(17,100) = and f(11,400) = .
4
8
1
1
1
÷ =
4
8
2
Therefore, 5700 years after 11,400, half of the carbon-14 is remaining.
t
(⁻ _
80,000 )
c. f = 2
_ _ _
Module 11
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Have students begin with the number 40 and take
half of it, then half of that result, then half of that, and
so on. Discuss with students when there might no
longer be anything to take half of. Students should
continue taking half until it is apparent to them that
they will never reach 0, although they will come
close. Discuss how a graph of ordered pairs (number
of times halved, result) would look.
© Houghton Mifflin Harcourt Publishing Company • ©Blaine Harrington III/
Alamy
b. The duration of 5700 years is referred to as the “half-life” of
carbon-14 because the amount of carbon-14 drops in half
5700 years after any starting point (not just t = 0 years).
Verify this property by comparing the amount of remaining
carbon-14 after 11,400 years and 17,100 years.
Lesson 1
EXTENSION ACTIVITY
A1_MTXESE353879_U5M11L1.indd 524
Have students make a table of values for uranium-234, similar to the table they
made in Part A of the Lesson Performance Task for carbon-14. Then have students
describe any patterns they see in the tables as they substitute values for t.
11/13/14 12:19 PM
Students will find that the values in the last two columns are exactly like the values
in the last two columns of the table for carbon-14.
Scoring Rubric
2 points: Student correctly solves the problem and explains his/her reasoning.
1 point: Student shows good understanding of the problem but does not fully
solve or explain his/her reasoning.
0 points: Student does not demonstrate understanding of the problem.
Understanding Rational Exponents and Radicals
524