math 10c - unit 2 workbook€¦ · · 2016-09-02numbers, radicals, and exponents. mathematics 10c...

Lesson 1: Number SetsApproximate Completion Time: 1 Day

Lesson 2: Primes, LCM, and GCFApproximate Completion Time: 2 Days

Lesson 3: Squares, Cubes, and RootsApproximate Completion Time: 2 Days

Lesson 4: RadicalsApproximate Completion Time: 2 Days

R

Q W

I N

Q

12 = 2 × 2 × 3

12

4 3

22

53 = 12552 = 25

am + n

Lesson 5: Exponents IApproximate Completion Time: 1 Day

Lesson 6: Exponents IIApproximate Completion Time: 2 Days

Mathematics 10CUnit

2Student Workbook

UNIT TWONumbers, Radicals, and Exponents

Mathematics 10CUnit

2Student Workbook

Complete this workbook by watching the videos on www.math10.ca.Work neatly and use proper mathematical form in your notes.

UNIT TWONumbers, Radicals, and Exponents

Numbers, Radicals, and ExponentsLESSON ONE - Number Sets

Lesson Notes

R

Q W

I N

Q

Introduction

Define each of the following sets of numbers andfill in the graphic organizer on the right.

a) Natural Numbers

b) Whole Numbers

c) Integers

d) Rational Numbers

e) Irrational Numbers

f) Real Numbers

www.math10.ca

Numbers, Radicals, and ExponentsLESSON ONE - Number SetsLesson Notes

R

Q W

I N

Q

Example 1 Determine which sets each number belongs to.In the graphic organizer, shade in the sets.

a) -4 b) 0 c) 1.273958... d) 7

e) 7.4 f) 4.93 g) h) π23

-

Example 2 For each statement, circle true or false.

a) All natural numbers are whole numbers. T F

b) All rational numbers are integers. T F

c) Some rational numbers are integers. T F

d) Some whole numbers are irrational numbers. T F

e) Rational numbers are real numbers, but irrational numbers are not. T F

www.math10.ca

R

Q W

I N

Q Numbers, Radicals, and ExponentsLESSON ONE - Number Sets

Lesson Notes

Example 3 Sort the following numbers as rational, irrational, or neither.You may use a calculator.

Rational Irrational Neither

www.math10.ca

Example 4 Order the numbers from least to greatest on a number line.You may use a calculator.

a)

b)

c)

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3-0.25-0.5-0.75-1-1.25-1.5-1.75-2-2.25-2.5-2.75-3

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3-0.25-0.5-0.75-1-1.25-1.5-1.75-2-2.25-2.5-2.75-3 3.25 3.5 3.75 4-3.25-3.5-3.75-4

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3-0.25-0.5-0.75-1-1.25-1.5-1.75-2-2.25-2.5-2.75-3

Numbers, Radicals, and ExponentsLESSON ONE - Number SetsLesson Notes

R

Q W

I N

Q

www.math10.ca

Numbers, Radicals, and ExponentsLESSON TWO - Primes, LCM, and GCF

Lesson Notesrise

run

12 = 2 × 2 × 3

12

4 3

22

Introduction

a) What is a prime number?

b) What is a composite number?

c) Why are 0 and 1 not considered prime numbers?

Prime Numbers, Least Common Multiple, and Greatest Common Factor.

www.math10.ca

Numbers, Radicals, and ExponentsLESSON TWO - Primes, LCM, and GCFLesson Notes

12 = 2 × 2 × 3

12

4 3

22

d) What is prime factorization? Find the prime factorization of 12.

e) What is the LCM? Find the LCM for 9 and 12 using two different methods.

f) What is the GCF? Find the GCF for 16 and 24 using two different methods.

www.math10.ca


Lesson Notes12 = 2 × 2 × 3

12

4 3

22

Example 1 Determine if each number is prime, composite, or neither.

a) 1 b) 14 c) 13 d) 0

Example 2 Find the least common multiple for each set of numbers.

a) 6, 8 b) 7, 14

c) 48, 180 d) 8, 9, 21

www.math10.ca

Example 3 Find the greatest common factor for each set of numbers.

a) 30, 42 b) 13, 39

c) 52, 78 d) 54, 81, 135


12 = 2 × 2 × 3

12

4 3

22

www.math10.ca


Lesson Notes12 = 2 × 2 × 3

12

4 3

22

Example 4

a) A fence is being constructed with posts that are 12 cm wide. A second fence is being constructed with posts that are 15 cm wide. If each fence is to be the same length, what is the shortest fencethat can be constructed?

Problem solving with LCM

b) Stephanie can run one lap around a track in 4 minutes. Lisa can run one lap in 6 minutes.If they start running at the same time, how long will it be until they complete a lap together?

c) There is a stack of rectangular tiles, with each tile having a length of 84 cm and a width of 63 cm. If some of these tiles are arranged into a square, what is the smallest side length the square can have?

www.math10.ca


12 = 2 × 2 × 3

12

4 3

22

Example 5

a) A fruit basket contains apples and oranges. Each basket will have the same quantity of apples, and the same quantity of oranges. If there are 10 apples and 15 oranges available, how many fruit baskets can be made? How many apples and oranges are in each basket?

b) There are 8 toonies and 20 loonies scattered on a table. If these coins are organized intogroups such that each group has the same quantity of toonies and the same quantity of loonies,what is the maximum number of groups that can be made? How many loonies and toonies are in each group?

Problem solving with GCF

c) A box of sugar cubes has a length of 156 mm, a width of 104 mm, and a height of 39 mm. What is the edge length of one sugar cube? Assume the box is completely full and the manufacturer uses sugar cubes with the largest possible volume.

SUGAR CUBES SUGAR CUBES

www.math10.ca

Numbers, Radicals, and ExponentsLESSON THREE - Squares, Cubes, and Roots

Lesson Notes

53 = 12552 = 25

Introductiona) What is a perfect square? Draw the first three perfect squares.

b) What is a perfect cube? Draw the first three perfect cubes.

Perfect Squares, Perfect Cubes, and Roots.

www.math10.ca


Lesson Notes

53 = 12552 = 25

c) Complete the table showing all perfect squares and perfect cubes up to 10. The first threeare completed for you.

d) What is a square root? Find the square root of 36.

e) What is a cube root? Find the cube root of 125.

Number Perfect Square Perfect Cube

1 12 = 1 13 = 1

2 22 = 4 23 = 8

3 32 = 9 33 = 27

i) Using a geometric square. ii) Using the formula A = s2

i) Using a geometric cube. ii) Using the formula V = s3

www.math10.ca


Lesson Notes

53 = 12552 = 25

Example 1 Evaluate each power, without using a calculator.

a) 32 b) (-3)2 c) -32

d) 33 e) (-3)3 f) -33

Example 2 Evaluate each expression, without using a calculator.

a) 2(2)3 b) -2(-4)2 c) 1 - 52

d) e) f )43

1 1

22 + 23

5(-2)3

-22

www.math10.ca


Lesson Notes

53 = 12552 = 25

Example 3 Evaluate each root using a calculator.

a) b) c) d)

Example 4 Evaluate each expression, without using a calculator.

e) What happens when you evaluate and ?Is there a pattern as to when you can evaluate the root of a negative number?

a) b)

c) d)

www.math10.ca


Lesson Notes

53 = 12552 = 25

Example 5 The area of Edmonton is 684 km2

a) If the shape of Edmonton is approximated to be a square, how wide is the city?

b) If the shape of Edmonton is approximated to be a circle, how wide is the city?

Example 6 The formula for the volume of a sphere is V = 43

πr3

a) If a sphere has a radius of 9 cm, what is the volume?

b) If a sphere has a volume of approximately 5000 cm3, what is the radius?

Edmonton

r = 9 cm

V = 5000 cm3

www.math10.ca


Lesson Notes

53 = 12552 = 25

Example 7

π = 2

9.8l

T

The amount of time, T, it takes for a pendulum to swing back and forth is called the period.

a) What is the period of the pendulum if the length, l, is 1.8 m?

b) What is the length of the pendulum if the period is 2.4 s?

The period of a pendulum can be calculated with the formula:

www.math10.ca


Lesson Notes

53 = 12552 = 25

Example 8 The total volume of gold mined throughout history is approximately 8340 m3.

a) If all the gold was collected, melted down, and recast as a cube, what would be the edge length?

b) If the density of gold is 19300 kg/m3, what is the mass of the cube?

c) In 2011, 1 kg of gold costs about $54 000. What is the value of all the gold ever extracted?

The density formula is density = mass

volume

www.math10.ca

This page is left blank intentionally for correct page alignment.

Numbers, Radicals, and ExponentsLESSON FOUR - Radicals

Lesson Notes

Introduction Understanding Radicals

a) Label each of the following parts of a radical.

3 8

b) What is the index of 5

c) What is the difference between an entire radical and a mixed radical?

?

d) Is it possible to write a radical without using the radical symbol ?

www.math10.ca

Numbers, Radicals, and ExponentsLESSON FOUR - RadicalsLesson Notes

Example 1 Convert each entire radical to a mixed radical.

a)

b)

c)

Prime Factorization Method Perfect Square Method

Prime Factorization Method Perfect Square Method

Prime Factorization Method Perfect Cube Method

www.math10.ca


Lesson Notes

Example 2 Convert each entire radical to a mixed radical using the method of your choice.

a) b)

c) d)

e) f)

www.math10.ca


Example 3 Convert each mixed radical to an entire radical.

a)

b)

c)

Reverse Factorization Method Perfect Square Method

Reverse Factorization Method Perfect Square Method

Reverse Factorization Method Perfect Cube Method

www.math10.ca


Lesson Notes

Example 4 Convert each mixed radical to an entire radical using the method of your choice.

a) b)

c) d)

Example 5 Estimate each radical and order them on a number line.

a)

b)

0 5 10

0 5 10

www.math10.ca


Example 6 Simplify each expression without using a calculator.

a) b)

c) d)

e)

www.math10.ca


Lesson Notes

Example 7 Write each power as a radical.

Example 8 Write each radical as a power.

a) b)

d) e)

c)

f)

a) b)

d) e)

c)

f)

www.math10.ca

Numbers, Radicals, and ExponentsLESSON FIVE - Exponents I

Lesson Notes

am + n

Introduction Exponent Laws I

a) Product of Powers

b) Quotient of Powers

c) Power of a Power

d) Power of a Product

e) Power of a Quotient

f) Exponent of Zero

General Rule:

General Rule:

General Rule:

General Rule:

General Rule:

General Rule:

www.math10.ca

Numbers, Radicals, and ExponentsLESSON FIVE - Exponents ILesson Notes

am + n

Example 1 Simplify each of the following expressions.

a) b)

c) d)

e) f)

www.math10.ca


Lesson Notes

am + n

Example 2

a) b)

c) d)

e) f)

Simplify each of the following expressions.

www.math10.ca

Numbers, Radicals, and ExponentsLESSON FIVE - Exponents ILesson Notes

am + n

Example 3

a) b)

c) d)

e) f)

Simplify each of the following expressions.

www.math10.ca


Lesson Notes

am + n

Example 4 For each of the following, find a value for m that satisfies the equation.

a) b)

c) d)

www.math10.ca

Numbers, Radicals, and ExponentsLESSON SIX - Exponents II

Lesson Notes

Introduction Exponent Laws II

a) Negative Exponents

b) Rational Exponents

General Rule:

General Rule:

www.math10.ca

Numbers, Radicals, and ExponentsLESSON SIX - Exponents IILesson Notes

Example 1

a) b)

c) d)

e) f)

Simplify each of the following expressions. Any variables in your final answer should be written with positive exponents.

www.math10.ca

a) b)

c) d)

e) f)


Lesson Notes

Example 2 Simplify. Any variables in your final answer should be written with positive exponents.

www.math10.ca

Example 3

a) b)

c) d)

Simplify each of the following expressions. Any variables in your final answer should be written with positive exponents.


www.math10.ca

a) b)

c) d)

Simplify. Any variables in your final answer should be written with positive exponents. Fractional exponents should be converted to a radical.


Lesson Notes

Example 4

www.math10.ca


Example 5

a) b)

c) d)

Simplify. Any variables in your final answer should be written with positive exponents. Fractional exponents should be converted to a radical.

www.math10.ca


Lesson Notes

Example 6

a) b)

c) d)

Write each of the following radical expressions with rational exponents and simplify.

www.math10.ca


Example 7

A culture of bacteria contains 5000 bacterium cells. This particular typeof bacteria doubles every 8 hours. If the amount of bacteria is representedby the letter A, and the elapsed time (in hours) is represented by the letter t,the formula used to find the amount of bacteria as time passes is:

a) How many bacteria will be in the culture in 8 hours?

b) How many bacteria will be in the culture in 16 hours?

c) How many bacteria were in the sample 8 hours ago?

www.math10.ca


Lesson Notes

Example 8

Over time, a sample of a radioactive isotope will lose its mass. The length of time for the sample to lose half of its mass is called the half-life of the isotope. Carbon-14 is a radioactive isotope commonly used to date archaeological finds. It has a half-life of 5730 years.

If the initial mass of a Carbon-14 sample is 88 g, the formula used to find the mass remaining as time passes is given by:

In this formula, A is the mass, and t is time (in years) since the mass of the sample was measured.

a) What will be the mass of the Carbon-14 sample in 2000 years?

b) What will be the mass of the Carbon-14 sample in 5730 years?

c) If the mass of the sample is measured 10000 years in the future, whatpercentage of the original mass remains?

www.math10.ca

www.math10.ca

Answer Key

Example 8: a) b) c) d) e) f)

Example 6: a) b) c) d) e)

Example 4: a) b) c) d)

Example 1: a) b) c)Example 2: a) b) c) d) e) f)Example 3: a) b) c)

Numbers, Radicals, and Exponents Lesson One: Number SetsIntroduction: a) The set of natural numbers (N) can be thought of as the counting numbers.b) The whole numbers (W) include all of the natural numbers plus one additional number - zero.c) The set of integers (I) includes negative numbers, zero, and positive numbers.d) The set of rational numbers (Q) includes all integers, plus terminating and repeating decimals.e) Irrational numbers (Q) are non-terminating and non-repeating decimals.f) Real numbers (R) includes all natural numbers, whole numbers, integers, rationals, and irrationals.Example 1: a) I, Q, R b) W, I, Q, R c) Q, R d) N W I Q R e) Q R f) Q R g) Q R h) Q R Example 2: a) true b) false c) true d) false e) false

Example 3: Rational: Irrational: Neither:

Example 4: a) b) c)

Numbers, Radicals, and Exponents Lesson Two: Primes, LCM, and GCFIntroduction: a) A prime number is a natural number that has exactly two distinct natural number factors: 1 and itself.b) A composite number is a natural number that has a positive factor other than one or itself.c) 0 is not a prime number because it has infinite factors. 1 is not a prime number because it has only one factor - itself.d) Prime Factorization is the process of breaking a composite number into its primes. 12 = 2 × 2 × 3e) The LCM is the smallest number that is a multiple of two given numbers. LCM of 9 & 12 is 36.f) The GCF is the largest natural number that will divide two given numbers without a remainder. GCF of 16 & 24 is 8.

Reals

RationalsIntegers

Wholes

Naturals

Irrationals

Example 1: a) neither b) composite c) prime d) neither Example 2: a) 24 b) 14 c) 720 d) 504 Example 3: a) 6 b) 13 c) 26 d) 27Example 4: a) 60 cm b) 12 minutes c) 252 cmExample 5: a) 5 baskets, with 3 oranges and 2 apples in each. b) 4 groups, with 5 loonies and 2 toonies in each. c) cube edge = 13 mm

Numbers, Radicals, and Exponents Lesson Three: Squares, Cubes, and RootsIntroduction:

b) A perfect cube is a number that can be expressed as the product of three equal factors. First three perfect cubes: 1, 8, 27

a) A perfect square is a number that can be expressed as the product of two equal factors. First three perfect squares: 1, 4, 9

Number Perfect Square Perfect Cube1 12 = 1 13 = 1

2 22 = 4 23 = 8

3 32 = 9 33 = 27

4

5

6

7

8

9

10

43 = 6442 = 16

53 = 12552 = 25

63 = 21662 = 36

73 = 34372 = 49

83 = 51282 = 64

93 = 72992 = 81

103 = 1000102 = 100

c)

d) A square root is one of two equal factors of a number.The square root of 36 is 6.

e) A cube root is one of three equal factors of a number.The cube root of 125 is 5.

Example 1: a) 9 b) 9 c) -9 d) 27 e) -27 f) -27 Example 2: a) 16 b) -32 c) -24 d) 1/64 e) 1/12 f) 10Example 3: a) 2.8284... b) error c) 2 d) -2 e) error, -1.5157... The odd root of a negative number can be calculated, but the even root of a negative number is not calculable.

Example 4: a) 20 b) 1/3 c) -1/4 d) 7/10Example 5: a) 26.2 km b) 29.5 kmExample 6: a) 3054 cm3 b) 10.61 cm

Example 7: a) 2.7 s b) 1.4 m Example 8: a) 20.28 m b) 160 962 000 kg c) 8.7 trillion dollars

Numbers, Radicals, and Exponents Lesson Four: RadicalsIntroduction:

3 8radical

radicand

indexradical symbol

a)

b) the index is 2

c) an entire radical does not have a coefficient, but a mixed radical does.

d) Yes. Radicals can be represented with fractional exponents.

Example 5: a) b)

Example 7: a) b) c)

d) e) f)

www.math10.ca

Answer KeyNumbers, Radicals, and Exponents Lesson Five: Exponents I

Introduction:

a) , ,

b) , ,

c) , ,

d) , ,

e) , ,

f) 1, ,1 a0 = 1




Example 4: a) 5 b) 3 c) 2 d) 7

Numbers, Radicals, and Exponents Lesson Six: Exponents IIIntroduction:

a) , , , , b) , , , ,

Example 1: a) b) c) d) e) f) Example 2: a) b) c) d) e) f)




Example 7: a) 10 000 bacteria b) 20 000 bacteria c) 2500 bacteria Example 8: a) 69 g b) 44 g c) 30%


math 10c - unit 2 workbook€¦ · · 2016-09-02numbers, radicals, and exponents. mathematics 10c...

Documents