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Mathematics 10C Student Workbook R Q W Q N I 2 Lesson 1: Number Sets Approximate Completion Time: 1 Day 12 4 2 3 12 = 2 × 2 × 3 Lesson 2: Primes, LCM, and GCF Approximate Completion Time: 2 Days 2 52 = 25 53 = 125 Unit Lesson 3: Squares, Cubes, and Roots Approximate Completion Time: 2 Days Lesson 4: Radicals Approximate Completion Time: 2 Days am + n Lesson 5: Exponents I Approximate Completion Time: 1 Day Lesson 6: Exponents II Approximate Completion Time: 2 Days UNIT TWO Numbers, Radicals, and Exponents Mathematics 10C Unit Student Workbook 2 Complete this workbook by watching the videos on www.math10.ca. Work neatly and use proper mathematical form in your notes. UNIT TWO Numbers, Radicals, and Exponents R Q I W Q Numbers, Radicals, and Exponents LESSON ONE - Number Sets Lesson Notes N Introduction Define each of the following sets of numbers and fill 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 Exponents LESSON ONE - Number Sets Lesson Notes Example 1 W Q b) 0 c) 1.273958... e) 7.4 f) 4.93 g) - 2 3 d) 7 h) π For each statement, circle true or false. a) All natural numbers are whole numbers. b) All rational numbers are integers. T c) Some rational numbers are integers. T F F T F d) Some whole numbers are irrational numbers. T F e) Rational numbers are real numbers, but irrational numbers are not. www.math10.ca Q N I Determine which sets each number belongs to. In the graphic organizer, shade in the sets. a) -4 Example 2 R T F R Q W N I Example 3 Q Numbers, Radicals, and Exponents LESSON ONE - Number Sets Lesson Notes Sort the following numbers as rational, irrational, or neither. You may use a calculator. Rational Irrational www.math10.ca Neither Numbers, Radicals, and Exponents LESSON ONE - Number Sets Lesson Notes R W Q N I Order the numbers from least to greatest on a number line. You may use a calculator. Example 4 a) -3 -2.75 -2.5 -2.25 -2 -1.75 -1.5 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 -3 -2.75 -2.5 -2.25 -2 -1.75 -1.5 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 b) c) -4 -3.75 -3.5 -3.25 -3 -2.75 -2.5 -2.25 -2 -1.75 -1.5 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 www.math10.ca 1.5 1.75 Q 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 Numbers, Radicals, and Exponents 12 4 2 3 12 = 2 × 2 rise ×3 2 Introduction LESSON TWO - Primes, LCM, and GCF Lesson Notes run Prime Numbers, Least Common Multiple, and Greatest Common Factor. a) What is a prime number? b) What is a composite number? c) Why are 0 and 1 not considered prime numbers? www.math10.ca Numbers, Radicals, and Exponents LESSON TWO - Primes, LCM, and GCF Lesson Notes 12 4 2 3 2 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 12 = 2 × 2 × 3 Numbers, Radicals, and Exponents 12 4 2 3 12 = 2 × 2 × 3 LESSON TWO - Primes, LCM, and GCF Lesson Notes 2 Example 1 a) 1 Example 2 Determine if each number is prime, composite, or neither. b) 14 c) 13 d) 0 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 Numbers, Radicals, and Exponents LESSON TWO - Primes, LCM, and GCF Lesson Notes Example 3 12 4 2 3 12 = 2 × 2 × 3 2 Find the greatest common factor for each set of numbers. a) 30, 42 b) 13, 39 c) 52, 78 d) 54, 81, 135 www.math10.ca Numbers, Radicals, and Exponents 12 4 2 3 12 = 2 × 2 × 3 LESSON TWO - Primes, LCM, and GCF Lesson Notes 2 Example 4 Problem solving with LCM 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 fence that can be constructed? 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 Numbers, Radicals, and Exponents LESSON TWO - Primes, LCM, and GCF Lesson Notes Example 5 12 4 2 3 12 = 2 × 2 × 3 2 Problem solving with GCF 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 into groups 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? 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. www.math10.ca SUGAR CU BES SUGAR CUBES 52 = 25 53 = 125 Numbers, Radicals, and Exponents LESSON THREE - Squares, Cubes, and Roots Lesson Notes Introduction Perfect Squares, Perfect Cubes, and Roots. a) What is a perfect square? Draw the first three perfect squares. b) What is a perfect cube? Draw the first three perfect cubes. www.math10.ca Numbers, Radicals, and Exponents 52 = 25 53 = 125 LESSON THREE - Squares, Cubes, and Roots Lesson Notes c) Complete the table showing all perfect squares and perfect cubes up to 10. The first three are completed for you. Number Perfect Square Perfect Cube 1 12 = 1 13 = 1 2 22 = 4 23 = 8 3 32 = 9 33 = 27 d) What is a square root? Find the square root of 36. i) Using a geometric square. ii) Using the formula A = s2 e) What is a cube root? Find the cube root of 125. i) Using a geometric cube. ii) Using the formula V = s3 www.math10.ca 52 = 25 Numbers, Radicals, and Exponents 53 = 125 LESSON THREE - Squares, Cubes, and Roots Lesson Notes 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 a) 2(2)3 d) 1 43 Evaluate each expression, without using a calculator. b) -2(-4)2 e) 1 22 + 2 3 www.math10.ca c) 1 - 52 f) 5(-2)3 -22 Numbers, Radicals, and Exponents 53 = 125 52 = 25 LESSON THREE - Squares, Cubes, and Roots Lesson Notes Example 3 a) Evaluate each root using a calculator. b) c) d) e) What happens when you evaluate and ? Is there a pattern as to when you can evaluate the root of a negative number? Example 4 Evaluate each expression, without using a calculator. a) b) c) d) www.math10.ca 52 = 25 53 = 125 Numbers, Radicals, and Exponents LESSON THREE - Squares, Cubes, and Roots Lesson Notes 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? Edmonton 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 = 4 πr3 3 a) If a sphere has a radius of 9 cm, what is the volume? r = 9 cm b) If a sphere has a volume of approximately 5000 cm3, what is the radius? V = 5000 cm3 www.math10.ca Numbers, Radicals, and Exponents 52 = 25 53 = 125 LESSON THREE - Squares, Cubes, and Roots Lesson Notes Example 7 The amount of time, T, it takes for a pendulum to swing back and forth is called the period. The period of a pendulum can be calculated with the formula: T = 2 π 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? www.math10.ca l 9.8 52 = 25 53 = 125 Numbers, Radicals, and Exponents LESSON THREE - Squares, Cubes, and Roots Lesson Notes 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? mass The density formula is density = volume c) In 2011, 1 kg of gold costs about $54 000. What is the value of all the gold ever extracted? www.math10.ca This page is left blank intentionally for correct page alignment. Numbers, Radicals, and Exponents LESSON 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 Exponents LESSON FOUR - Radicals Lesson Notes Example 1 Convert each entire radical to a mixed radical. a) Prime Factorization Method Perfect Square Method b) Prime Factorization Method Perfect Square Method c) Prime Factorization Method Perfect Cube Method www.math10.ca Numbers, Radicals, and Exponents LESSON FOUR - Radicals 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 Numbers, Radicals, and Exponents LESSON FOUR - Radicals Lesson Notes Example 3 Convert each mixed radical to an entire radical. a) Reverse Factorization Method Perfect Square Method b) Reverse Factorization Method Perfect Square Method c) Reverse Factorization Method Perfect Cube Method www.math10.ca Numbers, Radicals, and Exponents LESSON FOUR - Radicals 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) 0 5 10 0 5 10 b) www.math10.ca Numbers, Radicals, and Exponents LESSON FOUR - Radicals Lesson Notes Example 6 Simplify each expression without using a calculator. a) b) c) d) e) www.math10.ca Numbers, Radicals, and Exponents LESSON FOUR - Radicals Lesson Notes Example 7 Write each power as a radical. a) b) c) d) e) f) Example 8 Write each radical as a power. a) b) c) d) e) f) www.math10.ca This page is left blank intentionally for correct page alignment. am + n Numbers, Radicals, and Exponents LESSON FIVE - Exponents I Lesson Notes Introduction Exponent Laws I a) Product of Powers General Rule: b) Quotient of Powers General Rule: c) Power of a Power General Rule: d) Power of a Product General Rule: e) Power of a Quotient General Rule: f) Exponent of Zero General Rule: www.math10.ca Numbers, Radicals, and Exponents LESSON FIVE - Exponents I Lesson Notes Example 1 Simplify each of the following expressions. a) b) c) d) e) f) www.math10.ca am + n am + n Numbers, Radicals, and Exponents LESSON FIVE - Exponents I Lesson Notes Example 2 Simplify each of the following expressions. a) b) c) d) e) f) www.math10.ca Numbers, Radicals, and Exponents LESSON FIVE - Exponents I Lesson Notes Example 3 Simplify each of the following expressions. a) b) c) d) e) f) www.math10.ca am + n am + n Numbers, Radicals, and Exponents LESSON FIVE - Exponents I Lesson Notes Example 4 For each of the following, find a value for m that satisfies the equation. a) b) c) d) www.math10.ca This page is left blank intentionally for correct page alignment. Numbers, Radicals, and Exponents LESSON SIX - Exponents II Lesson Notes Introduction Exponent Laws II a) Negative Exponents General Rule: b) Rational Exponents General Rule: www.math10.ca Numbers, Radicals, and Exponents LESSON SIX - Exponents II Lesson Notes Example 1 Simplify each of the following expressions. Any variables in your final answer should be written with positive exponents. a) b) c) d) e) f) www.math10.ca Numbers, Radicals, and Exponents LESSON SIX - Exponents II Lesson Notes Example 2 Simplify. Any variables in your final answer should be written with positive exponents. a) b) c) d) e) f) www.math10.ca Numbers, Radicals, and Exponents LESSON SIX - Exponents II Lesson Notes Example 3 Simplify each of the following expressions. Any variables in your final answer should be written with positive exponents. a) b) c) d) www.math10.ca Numbers, Radicals, and Exponents LESSON SIX - Exponents II Lesson Notes Example 4 Simplify. Any variables in your final answer should be written with positive exponents. Fractional exponents should be converted to a radical. a) b) c) d) www.math10.ca Numbers, Radicals, and Exponents LESSON SIX - Exponents II Lesson Notes Example 5 Simplify. Any variables in your final answer should be written with positive exponents. Fractional exponents should be converted to a radical. a) b) c) d) www.math10.ca Numbers, Radicals, and Exponents LESSON SIX - Exponents II Lesson Notes Example 6 Write each of the following radical expressions with rational exponents and simplify. a) b) c) d) www.math10.ca Numbers, Radicals, and Exponents LESSON SIX - Exponents II Lesson Notes Example 7 A culture of bacteria contains 5000 bacterium cells. This particular type of bacteria doubles every 8 hours. If the amount of bacteria is represented by 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 Numbers, Radicals, and Exponents LESSON SIX - Exponents II 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, what percentage of the original mass remains? www.math10.ca This page is left blank intentionally for correct page alignment. Answer Key Numbers, Radicals, and Exponents Lesson One: Number Sets Reals Rationals Integers Wholes Introduction: 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: Example 4: a) Naturals Irrationals Neither: b) c) Numbers, Radicals, and Exponents Lesson Two: Primes, LCM, and GCF Introduction: 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 × 3 e) 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. 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) 27 Example 4: a) 60 cm b) 12 minutes c) 252 cm Example 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 Roots Introduction: 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 d) A square root is one of two equal factors of a number. The square root of 36 is 6. c) 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 e) A cube root is one of three equal factors of a number. The cube root of 125 is 5. Number Perfect Square Perfect Cube 1 12 = 1 13 = 1 2 22 = 4 23 = 8 3 32 = 9 33 = 27 4 42 = 16 43 = 64 5 52 = 25 53 = 125 6 62 = 36 63 = 216 7 72 = 49 73 = 343 8 82 = 64 83 = 512 9 92 = 81 93 = 729 10 102 = 100 103 = 1000 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) 10 Example 3: a) 2.8284... b) error c) 2 d) -2 e) error, -1.5157... Example 4: a) 20 b) 1/3 c) -1/4 d) 7/10 The odd root of a negative number can be calculated, Example 5: a) 26.2 km b) 29.5 km but the even root of a negative number is not calculable. Example 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: Radicals Introduction: a) index 3 radical symbol radicand 8 radical Example Example Example Example 1: 2: 3: 4: a) a) a) a) b) b) b) b) c) c) c) c) Example 5: a) Example 6: a) b) c) c) an entire radical does not have a coefficient, but a mixed radical does. Example 7: a) b) c) d) Example 8: a) e) f) d) b) b) the index is 2 d) Yes. Radicals can be represented with fractional exponents. d) d) e) b) e) f) c) d) e) www.math10.ca f) Answer Key Numbers, Radicals, and Exponents Lesson Five: Exponents I Introduction: a) , , b) , c) , d) b) c) d) e) f) Example 2: a) b) c) d) e) f) Example 3: a) b) c) d) e) f) Example 4: a) 5 b) 3 c) 2 d) 7 , , e) f) , Example 1: a) , , 1, , 1, a0 = 1 Numbers, Radicals, and Exponents Lesson Six: Exponents II Introduction: a) , Example 1: a) Example 3: a) Example 5: a) , , , b) c) b) b) d) e) c) b) Example 7: a) 10 000 bacteria c) , f) d) d) , , , Example 2: a) b) Example 4: a) Example 6: a) b) 20 000 bacteria c) 2500 bacteria c) b) b) c) c) Example 8: a) 69 g www.math10.ca d) e) d) d) b) 44 g c) 30% f)