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Chapter 3: Factors, Roots, and Powers Section 3.1
50
Chapter 3: Factors, Roots, and Powers
Section 3.1: Factors and Multiples of Whole Numbers
Terminology:
Prime Numbers:
Any natural number that has exactly two factors, 1 and the number itself.
Prime Numbers Include: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, …
Composite Numbers:
Any natural number that is not a prime number.
Prime Factorization:
The prime factorization of a number is the number written as a product of its
prime factors.
Determining the Prime Factors of a Whole Number
Ex1. Write the prime factorization of each
(a) 150 (b) 720
(c) 1250 (d) 3300
Chapter 3: Factors, Roots, and Powers Section 3.1
51
(e) 2646 (f) 1440
Terminology:
Greatest Common Factor:
The greatest common factor of two or more numbers is the greatest factor the
numbers have in common.
Determining the Greatest Common Factor
Ex1. Determine the greatest common factor of each set of numbers.
(a) 138 and 198
(b) 126 and 144
(c) 175 and 325
Chapter 3: Factors, Roots, and Powers Section 3.1
52
Terminology:
Least Common Multiple:
The least common multiple of two or more numbers is the least number that is
divisible by each number.
Determining the Least Common Multiple
Determine the least common multiple of each set of numbers.
(a) 30 and 45
(b) 18 and 40
Chapter 3: Factors, Roots, and Powers Section 3.1
53
(c) 18, 20, and 30
(d) 28, 42, and 63
Practice Problems
5,7,8,9,10,11,15 pg 140
Chapter 3: Factors, Roots, and Powers Section 3.2
54
Section 3.2: Perfect Squares, Perfect Cubes, and Roots
Terminology:
Perfect Squares:
A perfect square is a number that is produced by multiplying a number by itself
(ie. squaring it).
Perfect Squares Include: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, …
Perfect Cubes:
A perfect cube is a number that is produced by multiplying a number by itself
three times (ie. cubing it).
Perfect Cubes Include: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, …
Radical:
Consists of a root symbol, an index, and a radicand. It can be rational (for
example √4) or irrational (for example √2).
The expression √xn
is a radical where n is the index and x is the radicand.
Square Root: A radical with an index of 2
(a square root often doesn’t have a visible index, this is the only radical
for which this is true).
Cube Root: A radical with an index of 3.
Determining the Square Root of a Whole Number
Ex. Determine the value of each.
(a) √1296 (b) √1764
NOTE: When dealing with square roots you must make two equal groups with
your prime factorization. The product of the factors in each group is the value of
the square root.
Chapter 3: Factors, Roots, and Powers Section 3.2
55
Determining the Cube Root of a Whole Number
Ex. Determine the value of each.
(a) √17283
(b) √27443
NOTE: When dealing with cube roots you must make three equal groups with your
prime factorization. The product of the factors in each group is the value of the square
root.
Using Roots to Solve a Problem
Ex1. A cube has a volume of 4913 in3. What is the surface area of the cube?
Ex2. A cube has volume 12167 ft3. What is the surface area of the cube?
Practice Problems
4,5,7,8 pg 146-47
Chapter 3: Factors, Roots, and Powers Section 3.3
56
Section 3.3: Mixed and Entire Radicals
Terminology:
Mixed Radical Form:
A radical of the form 𝑎 √𝑥𝑛
such as 4√5, 2√183
, 3√24
.
A mixed radical is often called a simplified radical.
Entire Radical Form:
A radical of the form √𝑥𝑛
such as √80, √1443
, √1624
.
A entire radical cannot have a coefficient.
Perfect Cubes Include: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, …
Expressing Radicals in Simplified Radical Form (Mixed Radical)
Determine the exact simplified radical form of each.
(a) √72 (b) √26
(c) √500 (d) √112
Chapter 3: Factors, Roots, and Powers Section 3.3
57
(e) √245 (f) √180
(f) √403
(g) √323
(h) √3243
(i) √1283
(j) √3244
(k) √324
Chapter 3: Factors, Roots, and Powers Section 3.3
58
Writing Mixed Radicals as Entire Radicals
Ex. Express each mixed radical as an entire radical.
(a) 4√3 (b) 3√23
(c) 2√25
(d) 7√5 (e) 2√43
(f) 2√34
Practice Questions
4,5,10,11,12,17,18,21 pg 218-19
Chapter 3: Factors, Roots, and Powers Section 3.5
59
Section 3.4: The Real Number System
Terminology:
Natural Numbers (𝑵):
All counting numbers starting with 1.
Include: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,…
Whole Numbers (𝑾):
All Natural Numbers as well as Zero.
Includes: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, …
Integers (𝑰):
All positive and negative Natural Numbers and Zero.
Includes: …, -3, -2, -1, 0, 1, 2, 3, …
Rational Numbers (𝑸):
Any number that can be written in the form of a fraction. This includes repeating and
terminating decimals.
Irrational Numbers (�̅�):
Any number that cannot be written in the form of a fraction. This includes any non-
repeating, non-terminating decimal.
Real Numbers (𝑹):
All natural, whole, integer, rational, or irrational number.
The real number system can be demonstrated on a diagram as shown below:
Each of the number systems exists as a subsystem of the one outside it.
ie. Any natural number is also a whole number, an integer, a rational number and also a
real number. However any Integer is also a rational and real number. Irrational
numbers are only subset of the real numbers.
N
W
I
Q
Q
Real Number System (R)
Chapter 3: Factors, Roots, and Powers Section 3.5
60
Classifying Numbers as Subsystems of the Real Number System
Classify each according to its subsets of the real number system. Display your answers
on a diagram.
2 , 𝜋 , √7 ,1
4 , −9 , √36 , 0 , −
5
9
Practice Problems
3,4,7 pg 211
Chapter 3: Factors, Roots, and Powers Section 3.5
61
Section 3.5: The Laws of Exponents
The Laws of Exponents:
There are several laws that we use when we deal with the use of exponents in
expressions and equations.
Law 1: Product of Powers
When we multiply two powers with the same base, we must add the values of the
exponents.
𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚+𝑛
Ex: Write each as a single power:
(a) 53 ∙ 54
(b) 32 ∙ 37
(c) (1
3)
5
∙ (1
3)
9
(d) (−7)2 ∙ (−7)4 ∙ (−7)7
Law 2: Quotient of Powers
When we divide two powers with the same base, we must subtract the values of the
exponents.
𝑎𝑚
𝑎𝑛= 𝑎𝑚−𝑛 𝑜𝑟 𝑎𝑚 ÷ 𝑎𝑛 = 𝑎𝑚−𝑛
Ex: Write each as a single power:
(a) 712
75
(b) (−11)8 ÷ (−11)2
(c) 3.55
3.5
(d) 25∙26
22∙27
Chapter 3: Factors, Roots, and Powers Section 3.5
62
Law 3: Power of a Power
When an exponent is applied to a power, we must multiply the powers of the two
exponents.
(𝑎𝑚)𝑛 = 𝑎𝑚 ∙ 𝑛
Ex: Write each as a single power:
(a) (62)4
(b) [(−11)8]5
(c) [(1
5)
4
]3
(d) [27∙23
24∙2]
3
Law 4: Power of a Product
When an exponent is applied to the product of two numbers, we use the distributive
property and apply that exponent to each of the factors.
(𝑎 ∙ 𝑏)𝑚 = 𝑎𝑚 ∙ 𝑏𝑚
Ex: Write each as a single power:
(a) (52 ∙ 63)4
(b) (26 ∙ 93)3
Chapter 3: Factors, Roots, and Powers Section 3.5
63
Law 5: Power of a Quotient
When an exponent is applied to the quotient of two numbers, we use the distributive
property and apply that exponent to both the numerator and denominator (divisor and
dividend).
(𝑎
𝑏)
𝑚
=𝑎𝑚
𝑏𝑚
Ex: Write each as a single power:
(a) (25
43)
2
(b) (47
72)5
Law 6: Power of Zero
When a base has an exponent of zero, the value of the power is one.
𝑎0 = 1
Ex: Write each as a single power:
(a) 50
(b) (−7)0
(c) (−1
5)
0
(d) (25 ∙26
29+ 74 − 1000000010)
0
Law 7: Power of One
When a base does not have an exponent, it actually has an exponent of one.
𝑎 = 𝑎1
Chapter 3: Factors, Roots, and Powers Section 3.5
64
Simplifying Exponential Expressions Involving Variables
Sometimes, in exponential expressions, we encounter bases that are variables in
addition to those that are constants like we have explored thus far.
We deal with variable bases in the same way that we would deal with numerical bases.
We just apply the laws of exponents in the same fashion as before, simplifying all bases
that share the same variable.
Ex. Simplify each expression using the laws of exponents.
(a) 𝑎2 ∙ 𝑏3 ∙ 𝑎5 ∙ 𝑏2
(b) (𝑥3𝑦5)3 ∙ (𝑥2𝑦)5
(c) 𝑥5𝑦4𝑧8
𝑥2𝑦3𝑧5
(d) 𝑚4 ∙ 𝑛7 ∙ 𝑛5 ∙ 𝑚3
(e) (𝑎4𝑏2)2 ∙ (𝑎𝑏3)4
(f) 𝑥3𝑦5𝑧7
𝑥3𝑦2𝑧4
Chapter 3: Factors, Roots, and Powers Section 3.5
65
(g) (𝑟5𝑠2𝑡4
𝑟2𝑠𝑡3)
2
∙ (𝑟2𝑠5𝑡3
𝑟2𝑠2𝑡)
3
(h) (5𝑥5𝑦3
2𝑥𝑦2 )3
NOTE: If the word evaluate is used in a
question a numerical answer is required.
(i) (𝑎2𝑏3𝑐5
𝑎𝑏𝑐3)
3
∙ (𝑎4𝑏4𝑐2
𝑎2𝑏)
2
(j) (7𝑥5𝑦3
3𝑥2𝑦)
2
Chapter 3: Factors, Roots, and Powers Section 3.5
66
Section 3.6: Negative Exponents and Reciprocals
Complete the following table:
Power Value
26
25
24
23
22
21
20
2−1
2−2
2−3
2−4
2−5
2−6
What do you notice about the values of each power when comparing its value for the
negative and corresponding positive exponent?
____________________________________________________________
____________________________________________________________
____________________________________________________________
____________________________________________________________
Chapter 3: Factors, Roots, and Powers Section 3.5
67
Law 8: Negative Exponents
When a power has negative exponent, it can be rewritten as the reciprocal of the base
with a positive exponent.
𝑎−𝑚 =1
𝑎𝑚 𝑜𝑟 (
𝑎
𝑏)
−𝑚
= (𝑏
𝑎)
𝑚
Ex: Evaluate:
(a) 3−2
(b) (−3
4)
−3
(c) 0.3−4
(d) 7−2
(e) (10
3)
−3
(f) (−1.5)−3
NOTE: When simplifying an exponential expression, the final
answer cannot contain a negative exponent!!!!
Chapter 3: Factors, Roots, and Powers Section 3.6
66
Negative Exponents in Exponential Expressions with Variables
Simplify each expression:
(a) 𝑎3𝑏2𝑐4
𝑎2𝑏4𝑐3
(b) 𝑥5𝑦2𝑧5
𝑥6𝑦5𝑧
(c) 𝑥3𝑦−3 ∙ 𝑥−2𝑦−2
(d) 𝑗4𝑘2 ∙ 𝑗−2𝑘−9
(e) 𝑎−2𝑏5𝑐−2
𝑎−6𝑏−2𝑐3
(f) 𝑥2𝑦−3𝑧−5
𝑥2𝑦4𝑧−2
Chapter 3: Factors, Roots, and Powers Section 3.6
67
(g) (3𝑥−2𝑦3)2(6𝑥𝑦−2)−2
(h) (4𝑎𝑏3)−2(2𝑎−2𝑏3)3
(i) (𝑥5𝑦2𝑧−3
𝑥2𝑦5𝑧2 )2
(𝑥2𝑦−2𝑧
𝑥−3𝑦−1𝑧3)−1
(j) (𝑗3𝑘−4𝑙7
𝑗5𝑘−2𝑙−3)−2
(𝑗−1𝑘𝑙5
𝑗5𝑘−2𝑙3)3
Chapter 3: Factors, Roots, and Powers Section 3.6
68
ERROR ANALYSIS
Identify any errors and write the correct solutions
(𝑥−4𝑦2
10𝑥2𝑦−4)
−2
= (10𝑥−6𝑦6)−2
= (10)−2𝑥12𝑦−12
= −20𝑥12𝑦−12
= −20𝑥12
𝑦12
ERRORS:
CORRECT SOLUTION:
Chapter 3: Factors, Roots, and Powers Section 3.7
68
Section 3.6: Rational/Fractional Exponents
Law 9: Rational Exponents
When a power has a rational exponent, its numerator represents the applied the
exponent and the denominator represents the index of the applied radical(root).
𝑎𝑚
𝑛⁄ = √𝑎𝑚𝑛 𝑜𝑟 𝑎
𝑚𝑛⁄ = ( √𝑎
𝑛)
𝑚
Writing an Expression in Radical and Exponential Form
A expression may be written in one of two ways when considering rational exponents. It
may be given in exponential form or radical form. You may be asked to convert between
these forms.
Ex. Express each in its equivalent radical form:
(a) 52
3 (b) 135
2 (c) 𝑥3
4
(d) 117
4 (e) 39
5 (d) 𝑧4
11
Ex. Express each as a single power
(a) √543 (b)(√7)
5 (c) √𝑥36
(e) (√94
)5 (e)√393
(d) (√𝑥)5
Chapter 3: Factors, Roots, and Powers Section 3.7
69
Evaluating Powers with Rational Exponents
Evaluate each power.
(Hint: Recommend that you apply the radical first then the exponent)
(a) 811
2 (b) (−64)2
3
(c) (0.0016)3
4 (d) (−32)0.4
(e) (4
9)
3
2 (f) 1000
1
3
(g) (−343)2
3 (h) 0.813
2
(i) (0.0625)0.75 (j) (27
8)
2
3
Chapter 3: Factors, Roots, and Powers Section 3.7
70
(k) (25
121)
−3
2 (l) (
625
256)
−1
4
Simplifying and Evaluating Rational Expressions
Simplify to a single power and Evaluate
(a) 0.3−3 ∙ 0.35 (b) 0.22 ∙ 0.2−7
(c) [(−3
2)
−4
]2
∙ [(−3
2)
2
]3
(d) [(−4
5)
2
]−3
÷ [(−4
5)
4
]−5
(e) (1.4)3(1.4)4
(1.4)−2 (f) (1.5−3)
−5
(1.5)5
(g) (7
23
713 ∙ 7
53
)
6
(h)9
54 ∙ 9
−14
934
Chapter 3: Factors, Roots, and Powers Section 3.7
71
Rational Exponents in Exponential Expressions with Variables
Simplify each expression:
(a) (𝑥−3
2𝑦2) (𝑥1
2𝑦−1) (b) (𝑥3𝑦−1
2) (𝑥−5𝑦3
2)
(c) (8𝑎3𝑏6)2
3 (d) (25𝑎4𝑏2)3
2
(e) 4𝑎−2𝑏
23
2𝑎2𝑏13
(f) 12𝑥−5𝑦
52
3𝑥12𝑦
−12
Chapter 3: Factors, Roots, and Powers Section 3.7
72
(g) (100𝑎
25𝑎5𝑏−
12
)
1
2 (h) (
50𝑥2𝑦4
2𝑥4𝑦7)
1
2
Error Analysis
Identify any error(s) in each solution and write the correct solution
(𝑟12 ∙ 𝑠−
32)
12
∙ (𝑟−14 ∙ 𝑠
12)
−1
= 𝑟1 ∙ 𝑠−1 ∙ 𝑟−54 ∙ 𝑠−
12
= 𝑟1−54 ∙ 𝑠−1−
12
= 𝑟−14 ∙ 𝑠−
32
=1
𝑟14 ∙ 𝑠
32
ERRORS:
CORRECT SOLUTION:
Practice Problems
4,8,10,11,16,17,19,21 pg 241-43