Download Lesson 6.2: Properties of Exponents and Radicals

Lesson 6.2
NYS COMMON CORE MATHEMATICS CURRICULUM
Name:
Class:
Teacher: Mr. Free
Date: 11/10/14
M3
ALGEBRA II
Lesson 6.2: Properties of Exponents and Radicals
Classwork
Opening Exercise
Write each exponent as a radical, and then use the definition and properties of radicals to write that expression as an
integer.
1
1
1
1
a.
72 ⋅ 72
b.
33 ⋅ 33 ⋅ 33
c.
122 ⋅ 32
d.
(643 )
1
1
1
1
1 2
Examples 1–3
𝑚
Write each expression in the form 𝑏 𝑛 for positive real numbers 𝑏 and integers 𝑚 and 𝑛 with 𝑛 > 0 by applying the
properties of radicals and the definition of 𝑛th root.
1.
1
1
𝑏 4 ∙ 𝑏4
Lesson 4:
Date:
Properties of Exponents and Radicals
11/10/14
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Lesson 6.2
NYS COMMON CORE MATHEMATICS CURRICULUM
Name:
Class:
1
4
1
3
2.
𝑏3 ∙ 𝑏3
3.
𝑏5 ∙ 𝑏4
Teacher: Mr. Free
Date: 11/10/14
M3
ALGEBRA II
Exercises 1–4
𝑚
Write each expression in the form 𝑏 𝑛 . If a numeric expression is a rational number, then write your answer without
exponents.
1.
2.
2
1
𝑏3 ∙ 𝑏2
2
1 3
5
−
(𝑏 )
Lesson 4:
Date:
Properties of Exponents and Radicals
11/10/14
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Lesson 6.2
NYS COMMON CORE MATHEMATICS CURRICULUM
Name:
Class:
1
Teacher: Mr. Free
Date: 11/10/14
M3
ALGEBRA II
3
3.
643 ∙ 642
4.
( 2)
3
93 2
4
Example 4
Rewrite the radical expression √48𝑥5 𝑦4 𝑧2 so no perfect square factors remain inside the radical.
Exercise 5
5.
If 𝑥 = 50, 𝑦 = 12, and 𝑧 = 3, 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.
√8𝑥3 𝑦2
Lesson 4:
Date:
Properties of Exponents and Radicals
11/10/14
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Lesson 6.2
NYS COMMON CORE MATHEMATICS CURRICULUM
Name:
b.
Class:
Teacher: Mr. Free
Date: 11/10/14
M3
ALGEBRA II
3
√54𝑦7 𝑧2
Exercise 6
6.
Order these numbers from smallest to largest. Explain your reasoning.
162.5
93.1
Lesson 4:
Date:
321.2
Properties of Exponents and Radicals
11/10/14
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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 6.2
NYS COMMON CORE MATHEMATICS CURRICULUM
Name:
Class:
Teacher: Mr. Free
Date: 11/10/14
M3
ALGEBRA II
Lesson Summary
The properties of exponents developed in Grade 8 for integer exponents extend to rational exponents.
1
1
1
That is, for any integers 𝑚, 𝑛, 𝑝, and 𝑞, with 𝑛 > 0 and 𝑞 > 0 and any real numbers 𝑎 and 𝑏 so that 𝑎𝑛 , 𝑏𝑛 , and 𝑏𝑞
are defined, we have the following properties of exponents:
𝑚
𝑝
𝑚
𝑛
𝑚+𝑝
𝑞
𝑏 𝑛 ∙ 𝑏𝑞 = 𝑏 𝑛
𝑚
𝑏 𝑛 = √𝑏
1 𝑛
( 𝑏𝑛 )
1
𝑛 𝑛
(𝑏 )
=𝑏
=𝑏
𝑚
𝑚
𝑚
(𝑎𝑏) 𝑛 = 𝑎 𝑛 ⋅ 𝑏 𝑛
𝑝
𝑚𝑝
𝑚 𝑞
(𝑏 𝑛 ) = 𝑏 𝑛𝑞
𝑏
−𝑚
𝑛
=
1
𝑚
𝑏𝑛
Problem Set
1.
2.
Evaluate each expression if 𝑎 = 27 and 𝑏 = 64.
a.
3
b.
(3 3√𝑎√𝑏)
c.
( 3√𝑎 + 2√𝑏)
d.
𝑎−3 + 𝑏2
e.
(𝑎
−23
∙ 𝑏2 )
f.
(𝑎
−23
− 18 𝑏2 )
√𝑎√𝑏
2
2
3
2
3 −1
3 −1
Rewrite each expression so that each term is in the form 𝑘𝑥 𝑛 where 𝑘 is a real number, 𝑥 is a positive real number
and 𝑛 is a rational number.
2
1
a.
𝑥 −3 ∙ 𝑥 3
b.
2𝑥 2 ∙ 4𝑥 −2
1
5
1
c.
10𝑥 3
2𝑥 2
1 −2
d.
(3𝑥 4 )
Lesson 4:
Date:
Properties of Exponents and Radicals
11/10/14
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Lesson 6.2
NYS COMMON CORE MATHEMATICS CURRICULUM
Name:
Class:
1
𝑥 2 (2𝑥 2 − )
f.
3 27
𝑥6
h.
i.
Date: 11/10/14
ALGEBRA II
4
e.
g.
Teacher: Mr. Free
M3
𝑥
√
3
3
3
√𝑥 ∙ √−8𝑥 2 ∙ √27𝑥 4
2𝑥 4 −𝑥 2 −3𝑥
√𝑥
√𝑥−2𝑥 −3
4𝑥 2
2
3.
Show that (√𝑥 + √𝑦) is not equal to 𝑥1 + 𝑦1 when 𝑥 = 9 and 𝑦 = 16.
4.
Show that (𝑥2 + 𝑦2 )
5.
From these numbers, select one that is (a) negative, (b) one that is irrational, (c) one that is not a real number, and
(d) one that is a perfect square:
1
1 −1
is not equal to
1
1
+
𝑥2
1
1
1
𝑦2
1
when 𝑥 = 9 and 𝑦 = 16.
1
1
1
2
1
1
1 2
32 ∙ 92 , 273 ∙ 1442 , 643 − 643 , and (4−2 − 42 )
𝑛
6.
Show that this expression 2𝑛 ∙ 4𝑛+1 ∙ (18) is equal to 4.
7.
Express each answer as a power of 10.
8.
a.
Multiply 10𝑛 by 10.
b.
Multiply √10 by 10𝑛 .
c.
Square 10𝑛 .
d.
Divide 100 ∙ 10𝑛 by 102𝑛 .
e.
Show that 10𝑛 = 11 ∙ 10𝑛 − 10𝑛+1
Rewrite each of the following radical expressions as an equivalent exponential expression in which each variable
occurs no more than once.
a.
b.
9.
√8𝑥2 𝑦
5
√96𝑥3 𝑦15 𝑧6
Use properties of exponents to find two integers that are upper and lower estimates of the value of 41.6 .
Lesson 4:
Date:
Properties of Exponents and Radicals
11/10/14
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 6.2
NYS COMMON CORE MATHEMATICS CURRICULUM
Name:
Class:
Teacher: Mr. Free
Date: 11/10/14
M3
ALGEBRA II
10. Use properties of exponents to find two integers that are upper and lower estimates of the value of 82.3 .
11. Kepler’s third law of planetary motion relates the average distance, 𝑎, of a planet from the Sun to the time 𝑡 it takes
the planet to complete one full orbit around the Sun according to the equation 𝑡2 = 𝑎3 . When the time, 𝑡, is
measured in Earth years, the distance, 𝑎, is measured in astronomical units (AU). (One AU is equal to the average
distance from the earth to the sun.)
a.
Find an equation for 𝑡 in terms of 𝑎 and an equation for 𝑎 in terms of 𝑡.
b.
Venus takes about 0.616 Earth years to orbit the Sun. What is its average distance from the Sun?
c.
Mercury is an average distance of 0.387 AU from the Sun. About how long is its orbit in Earth years?
Lesson 4:
Date:
Properties of Exponents and Radicals
11/10/14
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.30
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.