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M7:LSN3 Existence and Uniqueness of Square and Cube Roots Activating Prior Knowledge β Write in exponential form. 1. π₯ β π₯ β π₯ β π₯ = π₯ 4 3. 2βπ₯β2βπ₯ 2. 3 β 3 β 3 = 22 π₯ 2 Estimate the value. 5. 17 β 4.1 = 33 4. z β π¦ β π§ β π¦ 6. 56 = π¦2π§ 2 β 7.5 Tie to LO Learning Objective Today, we will solve simple equations that require us to find the square or cube root of a number. CFU M7:LSN3 Module Page 11 Existence and Uniqueness of Square and Cube Roots Concept Development β Opening Exercise (8 mins.) Opening - The numbers in each column are related. Your goal is to determine how they are related, determine which numbers belong in the blank parts of the columns, and write an explanation for how you know the numbers belong there. Find the Rule Part 1 π π π π Find the Rule Part 2 π π π π ππ ππ π πππ ππ ππ πππ π πππ ππ ππ ππ ππ ππ ππ π πππ π π, πππ π π π CFU M7:LSN3 Existence and Uniqueness of Square and Cube Roots Module Page 11 Concept Development β Opening Discussion (9 mins.) For Find the Rule Part 1, how were you able to determine which number belonged in the blank? When given the number π in the left column, how did we Find the Rule Part 1 know the number that belonged to the right? ππ ππ ππ π ππ ππ ππ ππ π π π ππ ππππ πππ πππ πππ π ππππ ππ ππ ππ ππ πππ ππ When given the number π in the right column, how did we know the number that belonged to the left? πππ πππ πππ π π ππ π ππ CFU M7:LSN3 Module Page 11 Existence and Uniqueness of Square and Cube Roots Concept Development β Opening Discussion (9 mins.) For Find the Rule Part 2, how were you able to determine which number belonged in the blank? When given the number π in the left column, how did we Find the Rule Part 2 note the number that belonged to the right? π π π π ππ ππ π ππ ππ π ππππππ ππ ππππππ ππππ π, πππ π ππ ππ ππππ π, πππ ππ πππ ππ ππ π π When given the number π in the right column, the notation is similar to the notation we used to denote the square root. Given the number π in the right column, we write 3 π on the left. Were you able to write more than one number in any of the blanks? Were there any blanks that could not be filled? π, πππ π, πππ ππ π π CFU M7:LSN3 Existence and Uniqueness of Square and Cube Roots Module Page 11 Concept Development β Opening Discussion (9 mins.) For Find the Rule Part 1, you were working with squared numbers and square roots. For Find the Rule Part 2, you were working with cubed numbers and cube roots. Find the Rule Part 1 Find the Rule Part 2 π π π π π π π π ππ ππ ππ π π ππ πππ π ππ πππ ππ π π πππ π π πππ π, πππ π ππ ππ πππ ππ π, πππ ππ πππ π πππ ππ π πππ ππ ππ π π π π Just like we have perfect squares there are also perfect cubes. For example, 27 is a perfect cube because it is the product of 33 . Tell your partner another perfect cub. π π, πππ ππ π CFU M7:LSN3 Existence and Uniqueness of Square and Cube Roots Concept Development In the past, we have determined the length of the missing side of a right triangle, π₯, when ππ = ππ. What is that value and how did you get the answer? The value of π₯ is 5 because π₯ 2 means π₯ β π₯. Since 5 × 5 = 25, π₯ must be 5. CFU M7:LSN3 Existence and Uniqueness of Square and Cube Roots Concept Development When we solve equations that contain roots, we do what we do for all properties of equality, that is, we apply the operation to both sides of the equal sign. In terms of solving a radical equation, if we assume π₯ is positive, then: π₯2 π₯2 π₯ π₯ = 25 = 25 = 25 =5 Explain the first step in solving this equation. In Math 1 you will learn how to solve equations of this form without using the square root symbol, which means the possible values for π₯ can be both 5 and β5 because 52 = 25 and β5 2 = 25, but for now we will only look for the positive solution(s) to our equations. CFU M7:LSN3 Existence and Uniqueness of Square and Cube Roots Concept Development Consider the equation π₯ 2 = 25β1 . What is another way to write 25β1 ? The number 25β1 1 is the same as . 25 Again, assuming that π₯ is positive, we can solve the equation as before: We know we are correct because 1 2 5 = 1 25 = 25β1 . π₯ 2 = 25β1 1 π₯2 = 25 π₯2 = π₯= 1 25 1 25 1 π₯= 5 CFU M7:LSN3 Existence and Uniqueness of Square and Cube Roots Concept Development π The symbol is called a radical. Then an equation that contains that symbol is referred to as a radical equation. So far we have only worked with square 2 roots (π = 2). Technically, we would denote a square root as , but it is understood that the symbol alone represents a square root. 3 When π = 3, then the symbol is used to denote the cube root of a 3 number. Since π₯ 3 = π₯ β π₯ β π₯, then the cube root of π₯ 3 is π₯, i.e., π₯ 3 = π₯. For what value of π₯ is the equation π₯ 3 = 8 true? 3 π₯3 π₯3 π₯ π₯ =8 3 = 8 3 = 8 =2 The πth root of a number is π denoted by . In the context of our learning, we will limit our work with radicals to square and cube roots. CFU M7:LSN3 Existence and Uniqueness of Square and Cube Roots Skill Development/Guided Practice β Ex #1-9 Module Pages 12-13 Independent Work (10 minutes) Find the positive value of π₯ that makes each equation true. Check your solution. 1. π₯ 2 = 169 a. Explain the first step in solving this equation. You need to take the square root of both sides of the equation. a. Solve the equation and check your answer. π₯ 2 = 169 π₯2 = 169 π₯ = 13 Check: 132 = 169 169 = 169 CFU M7:LSN3 Existence and Uniqueness of Square and Cube Roots Skill Development/Guided Practice β Ex #1-9 Module Pages 12-13 Independent Work (10 minutes) 2. A square-shaped park has an area of 324 ft2. What are the dimensions of the park? Write and solve an equation. π΄ = π 2 π = 18 ft 324 = π 2 3. 625 = π₯ 2 625 = π₯2 π₯ = 25 4. A cube has a volume of 27 in3. What is the measure of one of its sides? Write and solve an equation. π£ = π 3 π = 3 in. 27 = π 3 3 27 = 3 π 3 CFU M7:LSN3 Existence and Uniqueness of Square and Cube Roots Skill Development/Guided Practice β Ex #1-9 Module Pages 12-13 Independent Work (10 minutes) 5. What positive value of π₯ makes the following equation true: π₯ 2 = 64? Explain. π₯ = 8 because 8 β 8 = 64 6. What positive value of π₯ makes the following equation true: π₯ 3 = 64? Explain. π₯ = 4 because 4 β 4 β 4 = 64 CFU M7:LSN3 Existence and Uniqueness of Square and Cube Roots Skill Development/Guided Practice β Ex #1-9 Module Pages 12-13 Independent Work (10 minutes) 7. π₯ 2 = 256β1 Find the positive value of x that makes the equation true. 1 π₯ = 256 2 π₯2 1 256 = π₯= 1 16 8. π₯ 3 = 343β1 Find the positive value of x that makes the equation true. π₯3 1 = 343 3 π₯3 3 = 1 343 1 π₯= 7 9. Is 6 a solution to the equation π₯ 2 β 4 = 5π₯? Explain why or why not. 62 β 4 = 5 6 36 β 4 = 30 Because 32 β 30, 6 is not a solution to the equation π₯ 2 β 4 = 5π₯. 32 β 30 CFU M7:LSN3 Module Page 14 Existence and Uniqueness of Square and Cube Roots Lesson Summary π The symbol is called a radical. Then an equation that contains that symbol is referred to as a radical equation. So far we have only worked with square roots (π = 2). Technically, we would 2 denote a positive square root as , but it is understood that the symbol alone represents a positive square root. 3 When π = 3, then the symbol is used to denote the cube root of a number. 3 Since π₯ 3 = π₯ β π₯ β π₯, then the cube root of π₯ 3 is π₯, i.e., π₯ 3 = π₯. The square or cube root of a positive number exists, and there can be only one positive square root or one cube root of the number. CFU Closure 1. What did we learn today? 2. Why is it important to you? 3. What is a cube root? How can you tell if you are looking for a square or a cube root Homework β Module page 14 Problem Set #1-9 Module Page 14 CFU Homework β Module page 14 Problem Set #1-9 Module Page 14 Find the positive value of π₯ that makes each equation true. Check your solution. 1. What positive value of π₯ makes the following equation true: π₯ 2 = 289? Explain. 2. A square shaped park has an area of 400 ft2. What are the dimensions of the park? Write and solve an equation. 3. A cube has a volume of 64 in3. What is the measure of one of its sides? Write and solve an equation. 4. What positive value of π₯ makes the following equation true: 125 = π₯ 3 ? Explain. CFU Homework β Module page 14 Problem Set #1-9 Module Page 14 5. π₯ 2 = 441β1 Find the positive value of x that makes the equation true. a. Explain the first step in solving this equation. b. Solve and check your solution. 6. π₯ 3 = 125β1 Find the positive value of x that makes the equation true. CFU Homework β Module page 14 Problem Set #1-9 Module Page 14 7. The area of a square is 196 in2. What is the length of one side of the square? Write and solve an equation, then check your solution. 8. The volume of a cube is 729 cm3. What is the length of one side of the cube? Write and solve an equation, then check your solution. 9. What positive value of π₯ would make the following equation true: 19 + π₯ 2 = 68? CFU
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