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Lesson 3: Properties of Exponents and Radicals
Unit 7
Lesson 3: Properties of Exponents and Radicals
Exercise 1.
Write each exponential expression as a radical expression, and then use the definition and properties of radicals to write
the resulting expression as an integer.
1
1
1
a. 72 ⋅ 72
1
2
c. 12 ⋅ 3
1
1
b. 33 ⋅ 33 ⋅ 33
1
1 2
3
1
2
d. (64 )
Exercise 2.
Show that you can get the same results using properties of exponents for each problem above.
The Properties of Exponents:
For any rational numbers (any number that can be written as a fraction) 𝑚 and 𝑛, and for any real numbers 𝑎 and 𝑏,
𝑏 𝑚 ⋅ 𝑏 𝑛 = 𝑏 𝑚+𝑛
𝑚
𝑛
𝑛
𝑏 𝑛 = √𝑏 𝑚 = ( √𝑏)
1 𝑚
𝑚
𝑚
(𝑏 𝑛 ) = 𝑏 𝑛
(𝑎 ⋅ 𝑏)𝑚 = 𝑎𝑚 ⋅ 𝑏 𝑚
(𝑏 𝑚 )𝑛 = 𝑏 𝑚⋅𝑛
1
𝑏 −𝑚 = 𝑚
𝑏
Lesson 3: Properties of Exponents and Radicals
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Lesson 3: Properties of Exponents and Radicals
Unit 7
Exercise 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 or exponents. If the expression is a rational number, then write your answer without exponents.
1
4
a. 𝑏 3 ⋅ 𝑏 3
2
1 3
5
−
c. (𝑏 )
3
93 2
e. ( 2)
4
1
3
b. 𝑏 5 ⋅ 𝑏 4
1
3
d. 643 ⋅ 642
28
f. ( 2 )
3
4
−
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Lesson 3: Properties of Exponents and Radicals
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Lesson 3: Properties of Exponents and Radicals
Unit 7
Exercise 4.
Rewrite the radical expression √48𝑥5 𝑦4 𝑧2 so that no perfect square factors remain inside the radical.
Exercise 5.
Use the definition of rational exponents and properties of exponents to rewrite each expression with rational exponents
containing as few fractions as possible. Then, evaluate each resulting expression for 𝑥 = 50, 𝑦 = 12, and 𝑧 = 3.
a. √8𝑥 3 𝑦 2
b.
3
√(54𝑦 7 𝑧 2
Exercise 6.
Order these numbers from largest to smallest without using a calculator. Explain your reasoning.
162.5
̅̅̅̅̅
83.666
321.2
b. 3𝜋
c. ( )√3
Exercise 7.
Evaluate the following irrational exponents:
1
a.
2√2
2
5
Lesson 3: Properties of Exponents and Radicals
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Lesson 3: Properties of Exponents and Radicals
Unit 7
Name:______________________
HOMEWORK
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.
𝑥
b.
1
2𝑥 2
c.
2.
2
3
−
a.
1
∙ 𝑥3
f.
5
∙ 4𝑥 −2
g.
1
10𝑥 3
h.
2𝑥 2
d.
1 −2
(3𝑥 4 )
e.
1
𝑥 2 (2𝑥 2
i.
3 27
√
3
𝑥6
3
3
√𝑥 ∙ √−8𝑥 2 ∙ √27𝑥 4
2𝑥 4 − 𝑥 2 − 3𝑥
√𝑥
√𝑥 − 2𝑥 −3
4𝑥 2
4
− )
𝑥
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:
1
1
1
2
1 1
1
1 2
−1
32 ∙ 92 , 273 ∙ 1442 , 643 − 643 , and (4 2 − 42 )
3.
𝑛
Show that the expression 2𝑛 ∙ 4𝑛+1 ∙ (1
) is equal to 4.
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Lesson 3: Properties of Exponents and Radicals
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