NYS COMMON CORE MATHEMATICS CURRICULUM
M3
Lesson 4
ALGEBRA II
Lesson 4: Properties of Exponents and Radicals
Date: 11/17/14
62
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Lesson 4: Properties of Exponents and Radicals
Student Outcomes
Students rewrite expressions involving radicals and rational exponents using the properties of exponents.
Lesson Notes
In Lesson 1, students reviewed the properties of exponents for integer exponents before establishing the meaning of the
th
root of a positive real number and how it can be expressed as a rational exponent in Lesson 3. In Lesson 4, students
extend properties of exponents that applied to expressions with integer exponents to expressions with rational
exponents. In each case, the notation
specifically indicates the principal root (e.g.,
is
, as opposed to
).
This lesson extends students’ thinking using the properties of radicals and the definitions from Lesson 3 so that they can
see why it makes sense that the properties of exponents hold for any rational exponents (N-RN.A.1). Examples and
exercises work to establish fluency with the properties of exponents when the exponents are rational numbers and
emphasize rewriting expressions and evaluating expressions using the properties of exponents and radicals (N-RN.A.2).
Classwork
Opening (2 minutes)
Students revisit the properties of square roots and cube roots studied in
Module 1 to remind them that we extended those to any
th
root in Lesson 3.
So, they are now ready to verify that the properties of exponents hold for
rational exponents.
Draw students’ attention to a chart posted prominently on the wall or to
their notebooks where the properties of exponents and radicals are
displayed, including those developed in Lesson 3.
Remind students of the description of exponential expressions of the form
, which they will be making use of throughout the lesson:
Let be any positive real number, and , be any integers with ; then
and
.
Scaffolding:
Throughout the lesson, remind students
of past properties of integer exponents
and radicals either through an anchor
chart posted on the wall or by writing
relevant properties as they come up.
Included is a short list of previous
properties used in this module.
For all real numbers , , and all
integers ,:
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
Lesson 4
ALGEBRA II
Lesson 4: Properties of Exponents and Radicals
Date: 11/17/14
63
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Opening Exercise (5 minutes)
These exercises briefly review content from Module 1 and the last lesson.
Opening Exercise
Write each exponent as a radical, and then use the definition and properties of radicals to write that expression as an
integer.
a.
b.
c.
d.
To transition from the Opening Exercise to Example 1, ask students to write the first two problems above in exponent
form. Then, ask them to discuss with a partner whether or not it would be true in general that
for
positive real numbers where , , , and are integers with and .
How could you write the
with rational exponents? How about
?
and
Based on these examples, is the exponent property
valid when and are rational
numbers? Explain how you know.
Since the exponents on the left side of each statement add up to the exponents on the right side, it
appears to be true. However, the right side exponent was always If we work with
and write it in exponent form, we would get (after noting
) that
. So, it appears
to be true in general. Note that examples alone do not prove that a mathematical statement is always
true.
In the rest of this lesson, we will make sense of these observations in general to extend the properties of exponents to
rational numbers by applying the definition of the root of and the properties of radicals introduced in Lesson 3.
MP.3
&
MP.8
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
Lesson 4
ALGEBRA II
Lesson 4: Properties of Exponents and Radicals
Date: 11/17/14
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Examples 1–3 (10 minutes)
In the previous lesson, we assumed that the exponent property
for positive real numbers and integers
and would also hold for rational exponents when the exponents were of the form
, where was a positive integer.
This example will help students see that the property below makes sense for any rational exponent:
, where , , , and are integers with and .
Perhaps model Example 1 below and then have students work with a partner on Example 2. Make sure students include
a justification for each step in the problem. When you get to Example 3, be sure to use the following discussion
questions to guide students.
How can we write these expressions using radicals?
In Lesson 3, we learned that
and
for positive real numbers and positive integers
and .
Which properties help us to write the expression as a single radical?
The property of radicals that states
for positive real numbers and and positive
integer , and the property of exponents that states
for positive real numbers and
integers and .
How do we rewrite this expression in exponent form?
In Lesson 3, we related radicals and rational exponents by
.
What makes Example 3 different from Examples 1 and 2?
The exponents have different denominators, so when we write the expression in radical form, the roots
are not the same, and we cannot apply the property that
.
Can you think of a way to rewrite the problem so it looks like the first two problems?
We can write the exponents as equivalent fractions with the same denominator.
Examples 1–3
Write each expression in the form
for positive real numbers and integers and with
by applying the properties of radicals and the definition of
th
root.
1.
By the definition of th root,
By the properties of radicals and properties of exponents
By the definition of
The rational number
is equal to
. Thus,
Scaffolding:
Throughout the lesson,
you can create parallel
problems to demonstrate
that these problems work
with numerical values as
well.
For example, in part (a)
substitute for .
In part (b), substitute a
perfect cube such as or
for
MP.7
&
MP.8
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
Lesson 4
ALGEBRA II
Lesson 4: Properties of Exponents and Radicals
Date: 11/17/14
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2.
By the definition of
and
By the properties of radicals and properties of exponents
By the definition of
Thus,
3.
Write the exponents as equivalent fractions with the same denominator.
Rewrite in radical form.
Rewrite as a single radical expression.
Rewrite in exponent form using the definition.
Thus,
Now add the exponents in each example. What is
?
?
?
,
, and
.
What do you notice about these sums and the value of the exponent when we rewrote each expression?
The sum of the exponents was equal to the exponent of the answer.
Based on these examples, particularly the last one, it seems reasonable to extend the properties of exponents to hold
when the exponents are any rational number. Thus, we can state the following property.
For any integers , , and, with and , and any real numbers so that
and
are defined,
.
Have students copy this property into their notes along with the ones listed below. You can also write these properties
on a piece of chart paper and display them in your classroom. These properties are listed in the lesson summary.
In a similar fashion, the other properties of exponents can be extended to hold for any rational exponents as well.
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
Lesson 4
ALGEBRA II
Lesson 4: Properties of Exponents and Radicals
Date: 11/17/14
66
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For any integers , , and, with and , and any real numbers and so that
,
, and
are defined,
At this point, you might have your class look at the opening exercise again and ask them which property could be used to
simplify each problem.
For advanced learners, a derivation of the property we explored in Example 1 is provided below.
Rewrite
and
as equivalent exponential expressions in which the exponents have the same denominator, and apply
the definition of the
as the
th
root.
By the definition of
and then using properties of algebra, we can rewrite the exponent to be
.
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M3
Lesson 4
ALGEBRA II
Lesson 4: Properties of Exponents and Radicals
Date: 11/17/14
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Exercises 1–4 (6 minutes)
Have students work with a partner or in small groups to complete these exercises.
Students are rewriting expressions with rational exponents using the properties
presented above. As students work, emphasize that we do not need to write these
expressions using radicals since we have just established that we believe that the
properties of exponents hold for rational numbers. In the last two exercises, students
will have to use their knowledge of radicals to rewrite the answers without
exponents.
Exercises 1–4
Write each expression in the form
. If a numeric expression is a rational number, then write
your answer without exponents.
1.
2.
3.
4.
Scaffolding:
When students get to
these exercises, you may
need to remind them that
it is often easier to rewrite
as
when you
are trying to evaluate
radical expressions.
For students who struggle
with arithmetic, you can
provide a scientific
calculator, but be sure to
encourage them to show
the steps using the
properties.
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
Lesson 4
ALGEBRA II
Lesson 4: Properties of Exponents and Radicals
Date: 11/17/14
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Example 4 (5 minutes)
We can rewrite radical expressions using properties of exponents. There are other methods for rewriting
radical expressions, but this example models using the properties of exponents. Often, textbooks and exams
give directions to simplify an expression, which is vague unless we specify what it means. We want students to
develop fluency in applying the properties, so the directions here say to rewrite in a specific fashion.
Example 4
Rewrite the radical expression
so that no perfect square factors remain inside the radical.
Although this process may seem drawn out, once it has been practiced, most of the steps can be internalized
and expressions are quickly rewritten using this technique.
Exercise 5 (5 minutes)
Students should work individually or in pairs on this exercise.
Exercise 5
5. If , , and , the following expressions are difficult to evaluate without using properties of radicals
or exponents (or a calculator). Use the definition of rational exponents and properties of exponents to rewrite each
expression in a form where it can be easily evaluated, and then use that exponential expression to find the value.
a.
Evaluating, we get
.
b.
Evaluating, we get
.
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
Lesson 4
ALGEBRA II
Lesson 4: Properties of Exponents and Radicals
Date: 11/17/14
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Lesson Summary
The properties of exponents developed in Grade 8 for integer exponents extend to rational exponents.
That is, for any integers , , , and, with and and any real numbers and so that
,
, and
are defined, we have the following properties of exponents:
Exercise 6 (5 minutes)
This exercise will remind students that rational numbers can be represented in decimal form and will give them a chance
to work on their numeracy skills. Students should work on this exercise with a partner or in their groups to encourage
dialogue and debate. Have a few students demonstrate their results to the entire class. There is more than one possible
approach, so when you debrief, try to share different approaches that show varied reasoning. Conclude with one or two
strong arguments. Students can confirm their reasoning using a calculator.
Exercise 6
6. Order these numbers from smallest to largest. Explain your reasoning.
is between and . We can rewrite
, which is
.
is between and . We can rewrite
, which is
.
is larger than
.
Thus,
is clearly the smallest number, but we need to determine if
is greater than or less than . To do
this, we know that
This means that
, and
, which is
greater than
Thus, the numbers in order from smallest to largest are
,
, and
.
Closing (2 minutes)
Have students summarize the definition and properties of rational exponents and any important ideas from the lesson
by creating a list of what they have learned so far about the properties of exponents and radicals. Circulate around the
classroom to informally assess understanding. Reinforce the properties of exponents listed below.
Exit Ticket (5 minutes)
MP.3
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
Lesson 4
ALGEBRA II
Lesson 4: Properties of Exponents and Radicals
Date: 11/17/14
70
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Name Date
Lesson 4: Properties of Exponents and Radicals
Exit Ticket
1. Find the exact value of
without using a calculator.
2. Justify that
using the properties of exponents in at least two different ways.
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
Lesson 4
ALGEBRA II
Lesson 4: Properties of Exponents and Radicals
Date: 11/17/14
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Exit Ticket Sample Solutions
1. Find the exact value of
without using a calculator.
2. Justify that
using the properties of exponents in at least two different ways.
Problem Set Sample Solutions
1. Evaluate each expression if and .
a.
b.
c.
d.
e.
f.
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
Lesson 4
ALGEBRA II
Lesson 4: Properties of Exponents and Radicals
Date: 11/17/14
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2. Rewrite each expression so that each term is in the form
, where is a real number, is a positive real number,
and is a rational number.
a.
b.
c.
d.
e.
f.
g.
h.
i.
3. Show that
is not equal to
when and .
When and , the two expressions are
and . The first expression simplifies to ,
and the second simplifies to . The two expressions are not equal.
4. Show that
is not equal to
when and .
When and , the two expressions are
and
. The first expression is
, and the
second one is
. The two expressions are not equal.
5. From these numbers, select (a) one that is negative, (b) one that is irrational, (c) one that is not a real number, and
(d) one that is a perfect square:
and
The first number,
, is irrational, the second number
is a perfect square, the third number,
, is negative, and the last number,
, is not a real number.
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
Lesson 4
ALGEBRA II
Lesson 4: Properties of Exponents and Radicals
Date: 11/17/14
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6. Show that the expression
is equal to .
7. Express each answer as a power of .
a. Multiply
by .
b. Multiply
by
.
c. Square
.
d. Divide
by
.
e. Show that
.
8. Rewrite each of the following radical expressions as an equivalent exponential expression in which each variable
occurs no more than once.
a.
b.
9. Use properties of exponents to find two integers that are upper and lower estimates of the value of
.
and
, so
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
Lesson 4
ALGEBRA II
Lesson 4: Properties of Exponents and Radicals
Date: 11/17/14
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10. Use properties of exponents to find two integers that are upper and lower estimates of the value of
.
and
, so
. Thus,
.
11. Kepler’s third law of planetary motion relates the average distance, , of a planet from the Sun to the time, , it
takes the planet to complete one full orbit around the Sun according to the equation
. When the time, , is
measured in Earth years, the distance, , is measured in astronomical units (AU). (One AU is equal to the average
distance from Earth to the Sun.)
a. Find an equation for in terms of and an equation for in terms of .
b. Venus takes about Earth years to orbit the Sun. What is its average distance from the Sun?
c. Mercury is an average distance of AU from the Sun. About how long is its orbit in Earth years?
