Pre-Algebra Chapter 3 Exponents and Roots Name: Period: Common Core State Standards CC.8.EE.1 - Know and apply the properties of integer exponents to generate equivalent numerical expressions. CC.8.EE.2 - Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 2 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that 2 is irrational. CC.8.EE.3 - Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is that the other. CC.8.EE.4 - Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities. Interpret scientific notation that has been generated by technology. CC.8.NS.1 - Understand informally that every number has a decimal expansion; the rational numbers are those with decimal expansions that terminate in 0s or eventually repeat. Know that other numbers are called irrational. CC.8.NS.2 - Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions. CC.8.G.6 - CC.8.G.7 - CC.8.G.8 - Explain a proof of the Pythagorean Theorem converse. Apply the Pythagorean Theorem to determine unknown side lengths in rigth triangles in real-world and mathematical problems in two and three dimensions. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. 1
Scope and Sequence Day 1 Lesson 3-1 Day 11 Lesson 3-6 Day 2 Lesson 3-2 Day 12 Lesson 3-6 Day 3 Lesson 3-2 Day 13 Lesson 3-7 Day 4 Lesson 3-3 Day 14 Lesson 3-7 Day 5 Lesson 3-4 Day 15 Extension Day 6 Lesson 3-4 Day 16 Lesson 3-8 Day 7 Tech Lab Day 17 Lesson 3-9 Day 8 Quiz Day 18 Review Day 1 Day 9 Lesson 3-5 Day 19 Review Day 2 Day 10 Lesson 3-5 Day 20 Test 2
IXL Modules SMART Score of 80 is required Due the day of the exam Lesson 1 8.F.6 Understanding negative exponents 8.F.7 Evaluate negative exponents Lesson 2 8.F.8 Multiplication with exponents 8.F.9 8.F.10 8.F.11 Division with exponents Multiplication and division with exponents Power rule Lesson 3 8.G.1 Convert between standard and scientific notation Lesson 4 8.G.3 Multiply numbers written in scientific notation 8.G.4 Divide numbers written in scientific notation Lesson 5 8.F.13 Square roots of perfect squares 8.F.14 8.F.16 Positive and negative square roots Relationship between squares and square roots Lesson 6 8.F.15 Estimate positive and negative square roots 8.F.18 8.F.20 Cube roots of perfect cubes Estimate cube roots Lesson 7 8.A.8 Classify Numbers Lesson 8 8.O.1 Pythagorean theorem: find the length of the hypotenuse 8.O.2 8.O.3 8.O.4 Pythagorean theorem: find the missing leg length Pythagorean theorem: find the perimeter Pythagorean theorem: word problems Lesson 9 8.O.5 Converse of the Pythagorean theorem: is it a right triangle? 8.P.4 Distance between two points 3
Lesson 3-1 Integer Exponents Warm-Up Examples: Using a Pattern to Simplify Negative Exponents Simplify. Write in decimal form. 10-2 10-1 10-8 10-9 4
Examples: Evaluating Negative Exponents Simplify. 5-3 (-10) -3 4-2 (-7) -4 Examples: Using the Order of Operations Evaluate 5 - (6-4) -3 + (-2) 0 Evaluate 3 + (7-4) -2 + (-8) 0 5
Lesson 3-2 Properties of Exponents Warm-Up Examples: Multiplying Powers with the Same Base Multiply. Write the product as one power. 6 6 3 n 5 n 7 2 5 2 24 24 4 4 2 4 4 x 2 x 3 x 5 y 2 41 2 41 7 6
Examples: Dividing Powers with the Same Base Divide. Write the quotient as one power. 7 5 x 10 7 3 x 9 9 9 9 2 e 10 e 5 7
Examples: Raising a Power to a Power Simplify. (5 4 ) 2 (6 7 ) 9 (17 2 ) -20 (3 3 ) 4 (4 8 ) 2 (13 4 8
Lesson 3-3 Scientific Notation Warm-Up Numbers written in scientific notation are written as factors. One factor is a number than or equal to 1 and than 10. The other factor is a of 10. Examples: Translating Scientific Notation to Standard Notation Write the number in standard notation. 1.35 x 10 5 2.7 x 10-3 2.01 x 10 4 2.87 x 10 9 1.9 x 10-5 9
Examples: Translating Standard Notation to Scientific Notation Write 0.00709 in scientific notation. Write 0.000811 in scientific notation. Examples: Application A pencil is 18.7 cm long. If you were to lay 10,000 pencils end-to-end, how many millimeters long would they be? Write the answer in scientific notation. An oil rig can hoist 2,400,000 pounds with its main derrick. It distributes the weight evenly between 8 cables. What s the weight that each cable can hold? Answer in scientific notation. A certain cell has a diameter of approximately 4.11 x 10-5 meters. A second cell has as a diameter of 1.5 x 10-5 meters. Which cell has a greater diameter? A certain cell has a diameter of approximately 5 x 10-3 meters. A second cell has as a diameter of 5.11 x 10-3 meters. Which cell has a greater diameter? 10
Lesson 3-4 Operating With Scientific Notation Warm-Up Examples: Division With Scientific Notation Pluto was demoted from planet to dwarf planet in 2006, in part because of its small size. The mass of Pluto is about 1.3 x 10 22 kg. About how many times smaller is Pluto s mass than Earth s mass? (Earth s mass is 5.97 x 10 24 kg) About how many times greater is the mass of Saturn than the mass of Earth? Write your answer in scientific notation. (Saturn s mass is 5.69 x 10 26 ) 11
Examples: Multiplication with Scientific Notation The mass of Venus is about 66 times the mass of the moon. What is the mass of the moon? Write your answer in scientific notation. (Mass of Venus is 4.87 x 10 24 ) The average mass of a grain of sand on a beach is about 1.5 x 10-5. There are about 6.1 x 10 12 grains of sand in a beach volleyball court. What is the mass of the grains of sand in the beach volleyball court? Examples: Addition and Subtraction with Scientific Notation The water in Mono Lake, CA, was used for residents of LA. As a result, the lake s volume, in acre-feet, dropped from 4.3 x 10 7 to 2.1 x 10 7 from 1941 to 1982. After becoming protected, the lake increased in volume by 5.0 x 10 6 acre-feet in the next 20 years. What was its volume in 2002? How much greater is the mass of Neptune than the mass of Earth? Write your answer in scientific notation. (Earth mass is 5.97 x 10 24, Neptune mass is 1.02 x 10 26 ) 12
Lesson 3-5 Equations, Tables and Graphs Warm-Up A number that when multiplied by to form a product is the square root of that product. Taking the square root of a number is the of squaring the number. Every positive number has two square roots, one and one. The radical symbol indicates the nonnegative or principal square root. They symbol - is used to indicate a negative square root. The numbers 16, 36, and 49 are examples of perfect squares. A perfect square is a number that has as its square roots. Other perfect squares include 1, 4, 9, 25 and 64. Examples: Finding the Positive and Negative Square Roots of a Number Find the two square roots of each number. A. 49 B. 100 C. 225 A. 25 B. 144 C. 289 13
Examples: Application A square window has an area of 169 square inches. How wide is the window? A square shaped kitchen table has an area of 16 square feet. Will it fit through a van door that has a 5 foot wide opening? Simplify the expression. Examples: Evaluating Expressions Involving Square Roots 3 36 + 7 25 3 16 + 4 2 25 + 4 2 18 + 4 1 14
Lesson 3-6 Estimating Square Roots Warm-Up Examples: Estimating Square Roots of Numbers The square root is between two integers. Name the integers. Explain your answer. 55-90 80-45 Examples: Application You want to sew a fringe on a square tablecloth with an area of 500 square inches. Calculate the length of each side of the tablecloth and the length of fringe you will need to the nearest tenth of an inch. 15
A tent was advertised in the newspaper as having an enclosed square area of 168 ft 2. What is the approximate length of the sides of the square area? Round your answer to the nearest foot. Examples: Approximating Square Roots to the Nearest Hundredth Approximate to the nearest hundredth. 141 240 Examples: Using a Calculator to Estimate the Value of a Square Root Use a calculator to find the square root. Round to the nearest tenth. 600 800 16
Lesson 3-7 The Real Numbers Warm-Up Irrational numbers can only be written as decimals that do terminate or repeat. If a whole number is not a square, then its square root is an irrational number. The set of real numbers consists of the set of numbers and the set of numbers. Write all names that apply to each number. 5-12.75 Examples: Classifying Real Numbers 9-35.9 17
Examples: Determining the Classification of All Numbers State if each number is rational, irrational or not a real number. 21 0 3 4 4 9 23 9 0 7 64 81 The Density Property of real numbers states that any two real numbers is another real number. This property is also true for rational numbers, but for whole numbers or integers. 18
Examples: Applying the Density Property of Real Numbers 2 3 Find a real number between 3 5 and 3 5. 3 4 Find a real number between 4 7 and 4 7. 19
Lesson 3-8 The Pythagorean Theorem Warm-Up Examples: Finding the Length of a Hypotenuse Find the length of the hypotenuse to the nearest hundredth. 20
Examples: Finding the Length of a Leg in a Right Triangle Solve for the unknown side in the right triangle to the nearest tenth. 21
Examples: Using the Pythagorean Theorem for Measurement Two airplanes leave the same airport at the same time. The first plane flies to a landing strip 350 miles south, while the other plane flies to an airport 725 miles west. How far apart are the two planes after they land? Two birds leave the same spot at the same time. The first bird flies to his nest 11 miles sout, while the other bird flies to his nest 7 miles west. How far apart are the two birds after they reach their nests? 22
Lesson 3-9 Applying the Pythagorean Theorem and Its Converse Warm-Up Examples: Marketing Application What is the diagonal length of the projector screen? What is the diagonal length of the projector screen? 23
Examples: Finding the Distance on the Coordinate Plane Find the distances between the points to the nearest tenth. J and K L and M 24
J and L K and M 25
Examples: Identifying a Right Triangle Tell whether the given side lengths form a right triangle. 9, 12, 15 8, 10, 13 5, 6, 9 8, 15, 17 26