Chapter 3: Factors, Roots, and Powers - Mr....

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


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(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


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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


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(c) 18, 20, and 30

(d) 28, 42, and 63

Practice Problems

5,7,8,9,10,11,15 pg 140


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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.


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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


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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


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(e) √245 (f) √180

(f) √403

(g) √323

(h) √3243

(i) √1283

(j) √3244

(k) √324


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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


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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)


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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


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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


(a) 712

75

(b) (−11)8 ÷ (−11)2

(c) 3.55

3.5

(d) 25∙26

22∙27


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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


(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


(a) (52 ∙ 63)4

(b) (26 ∙ 93)3


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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


(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


(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


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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


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(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


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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?

____________________________________________________________

____________________________________________________________

____________________________________________________________

____________________________________________________________


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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!!!!


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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


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(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


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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:


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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


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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


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(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


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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


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(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

Chapter 3: Factors, Roots, and Powers - Mr....

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