Lesson 3: Existence and Uniqueness of Square Roots and Cube Roots

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 8•7

Student Outcomes . Students know that the positive square root and the cube root exist for all positive numbers and both a square root of a number and a cube root of a number are unique. . Students solve simple equations that require them to find the square root or cube root of a number.

Lesson Notes This lesson has two options for showing the existence and uniqueness of positive square roots and cube roots. Each option has an Opening Exercise and a Discussion that follows. The first option has students explore facts about numbers on a number line, leading to an understanding of the trichotomy law, followed by a discussion of how the law applies to squares of numbers, which should give students a better understanding of what square roots and cube roots are and how they are unique. The second option explores numbers and their squares via a Find the Rule exercise, followed by a discussion that explores how square roots and cube roots are unique. The first option includes a discussion of the basic inequality property, a property referred to in subsequent lessons. The basic inequality property states that if 푥, 푦, 푤, and 푧 are positive numbers so that 푥 < 푦 and 푤 < 푧, then 푥푤 < 푦푧. Further, if 푥 = 푤 and 푦 = 푧, when 푥 < 푦, then 푥2 < 푦2. Once the first or second option is completed, the lesson continues with a discussion of how to solve equations using square roots. Throughout this and subsequent lessons, we ask students to find only the positive values of 푥 that satisfy an equation containing squared values of 푥. The reason is that students will not learn how to solve these problems completely until Algebra I. Consider the following equation and solution below: 4푥2 + 12푥 + 9 = 49 (2푥 + 3)2 = 49 √(2푥 + 3)2 = √49 2푥 + 3 = ±7 2푥 + 3 = 7 or 2푥 + 3 = −7 푥 = 2 푥 = −5 At this point, students have not learned how to factor quadratics and will solve all equations using the square root symbol, which means students are only responsible for finding the positive solution(s) to an equation.

Classwork Opening (4 minutes): Option 1 Ask students the following to prepare for the discussion that follows. . Considering only the positive integers, if 푥2 = 4, what must 푥 be to make the statement a true number sentence? Could 푥 be any other number?  푥 is the number 2 and cannot be any other number.

Lesson 3: Existence and Uniqueness of Square Roots and Cube Roots 36

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 8•7

. Can you find a pair of positive numbers 푐 and 푑 so that 푐 < 푑 and 푐2 < 푑2?  Yes! Answers will vary. . Can you find a pair of positive numbers 푐 and 푑 so that 푐 < 푑 and 푐2 = 푑2?  No . Can you find a pair of positive numbers 푐 and 푑 so that 푐 < 푑 and 푐2 > 푑2?  No

Discussion (9 minutes): Option 1 (An alternative discussion is provided below.) Once this discussion is complete, continue with the discussion that begins on page 40. . We will soon be solving equations that include roots. For this reason, we want to be sure that the answer we get when we simplify is correct. Specifically, we Scaffolding: want to be sure that we can get an answer, that it exists, that the answer we get The number line can help is correct, and that it is unique to the given situation. students make sense of the . To this end, existence requires us to show that given a positive number 푏 and a trichotomy law. Give students positive integer 푛, there is one and only one positive number 푐, so that when two numbers, and ask them 푐푛 = 푏, we say “푐 is the positive 푛th root of 푏.” When 푛 = 2, we say “푐 is the which of the three possibilities positive square root of 푏” and when 푛 = 3, we say “푐 is the positive cube root of for those numbers is true. In 푏.” Uniqueness requires us to show that given two positive numbers 푐 and 푑 each case, only one of three is and 푛 = 2: If 푐2 = 푑2, then 푐 = 푑. This statement implies uniqueness because true for any pair of numbers. both 푐 and 푑 are the positive square root of 푏, that is, 푐2 = 푏 and 푑2 = 푏; since For example, given the 푐2 = 푑2, then 푐 = 푑. Similarly, when 푛 = 3, if 푐3 = 푑3, then 푐 = 푑. The numbers 푐 = 2 and 푑 = 3, reasoning is the same since both 푐 and 푑 are the positive cube root of 푏, that is, which of the following is true: 푐3 = 푏 and 푑3 = 푏; since 푐3 = 푑3, then 푐 = 푑. Showing uniqueness also shows 푐 = 푑, 푐 < 푑, or 푐 > 푑? existence, so we focus on proving the uniqueness of square and cube roots. MP.3 . To show that 푐 = 푑, we use the trichotomy law. The trichotomy law states that given two numbers 푐 and 푑, one and only one of the following three possibilities is true. (i) 푐 = 푑 (ii) 푐 < 푑 (iii) 푐 > 푑 We show 푐 = 푑 by showing that 푐 < 푑 and 푐 > 푑 cannot be true. . If 푥, 푦, 푤, and 푧 are positive numbers so that 푥 < 푦 and 푤 < 푧, is it true that 푥푤 < 푦푧? Explain.  Yes, it is true that 푥푤 < 푦푧. Since all of the numbers are positive and both 푥 and 푤 are less than 푦 and 푧, respectively, then their product must also be less. For example, since 3 < 4 and 5 < 6, then 3 × 5 < 4 × 6. . This basic inequality property can also be explained in terms of areas of a rectangle. The picture on the right clearly shows that when 푥 < 푦 and 푤 < 푧, then 푥푤 < 푦푧.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 8•7

. We use this fact to show that 푐 < 푑 and 푐 > 푑 cannot be true. We begin with 푐푛 < 푑푛 when 푛 = 2. By the basic inequality, 푐2 < 푑2. Now, we look at the case where 푛 = 3. We use the fact that 푐2 < 푑2 to show 푐3 < 푑3. What can we do to show 푐3 < 푑3?  We can multiply 푐2 by 푐 and 푑2 by 푑. The basic inequality guarantees that since 푐 < 푑 and 푐2 < 푑2, that 푐2 × 푐 < 푑2 × 푑, which is the same as 푐3 < 푑3. . Using 푐3 < 푑3, how can we show 푐4 < 푑4? 3 3 3 3 MP.3  We can multiply 푐 by 푐 and 푑 by 푑. The basic inequality guarantees that since 푐 < 푑 and 푐 < 푑 , 푐3 × 푐 < 푑3 × 푑, which is the same as 푐4 < 푑4. . We can use the same reasoning for any positive integer 푛. We can use similar reasoning to show that if 푐 > 푑, then 푐푛 > 푑푛 for any positive integer 푛. . Recall that we are trying to show that if 푐푛 = 푑푛, then 푐 = 푑 for 푛 = 2 or 푛 = 3. If we assume that 푐 < 푑, then we know that 푐푛 < 푑푛, which contradicts our hypothesis of 푐푛 = 푑푛. By the same reasoning, if 푐 > 푑, then 푐푛 > 푑푛, which is also a contradiction of the hypothesis. By the trichotomy law, the only possibility left is that 푐 = 푑. Therefore, we have shown that the square root or cube root of a number is unique and also exists.

Opening (6 minutes): Option 2 Begin by having students find the rule given numbers in two columns. The goal is for students to see the relationship between the square of a number and its square root and the cube of a number and its cube root. Students have to figure out the rule, find the missing values in the columns, and then explain their reasoning. Provide time for students to do this independently. If necessary, allow students to work in pairs. . 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.

Opening

Find the Rule Part 1

ퟏ ퟏ ퟐ ퟒ ퟑ ퟗ MP.8 ퟗ ퟖퟏ ퟏퟏ ퟏퟐퟏ ퟏퟓ ퟐퟐퟓ ퟕ ퟒퟗ ퟏퟎ ퟏퟎퟎ ퟏퟐ ퟏퟒퟒ ퟏퟑ ퟏퟔퟗ 풎 풎ퟐ √풏 풏

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 8•7

Note: Students will not know how to write the cube root of a number using the proper notation, but this activity will be a good way to launch into the discussion below.

Find the Rule Part 2

ퟏ ퟏ ퟐ ퟖ ퟑ ퟐퟕ ퟓ ퟏퟐퟓ MP.8 ퟔ ퟐퟏퟔ ퟏퟏ ퟏ, ퟑퟑퟏ ퟒ ퟔퟒ ퟏퟎ ퟏ, ퟎퟎퟎ ퟕ ퟑퟒퟑ ퟏퟒ ퟐ, ퟕퟒퟒ 풑 풑ퟑ ퟑ√풒 풒

Discussion (7 minutes): Option 2 Once the Find the Rule exercise is finished, use the discussion points below; then, continue with the Discussion that follows on page 39. . For Find the Rule Part 1, how were you able to determine which number belonged in the blank?  To find the numbers that belonged in the blanks in the right column, I had to square the number in the left column. To find the numbers that belonged in the left column, I had to take the square root of the number in the right column. . When given the number 푚 in the left column, how did we know the number that belonged in the right column?  Given 푚 on the left, the number that belonged on the right was 푚2. . When given the number 푛 in the right column, how did we know the number that belonged in the left column?  Given 푛 on the right, the number that belonged on the left was √푛. . For Find the Rule Part 2, how were you able to determine which number belonged in the blank?  To find the number that belonged in the blank in the right column, I had to multiply the number in the left column by itself three times. To find the number that belonged in the left column, I had to figure out which number multiplied by itself three times equaled the number that was in the right column. . When given the number 푝 in the left column, how did we note the number that belonged in the right column?  Given 푝 on the left, the number that belonged on the right was 푝3. . When given the number 푞 in the right column, the notation we use to denote the number that belongs in the left column is similar to the notation we used to denote the square root. Given the number 푞 in the right column, we write 3√푞 in the left column. The 3 in the notation shows that we must find the number that multiplied by itself 3 times is equal to 푞.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 8•7

. Were you able to write more than one number in any of the blanks?  No, there was only one number that worked. . Were there any blanks that could not be filled?  No, in each case there was a number that worked. . 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. 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. For Find the Rule Part 2, you cubed the number on the left to fill the blank on the right and took the cube root of the number on the right to fill the blank on the left. . We could extend the Find the Rule exercise to include an infinite number of rows, and in each case, we would be able to fill the blanks. Therefore, we can say that positive square roots and cube roots exist. Because only one number worked in each of the blanks, we can say that the positive roots are unique. . We must learn about square roots and cube roots to solve equations. The properties of equality allow us to add, subtract, multiply, and divide the same number to both sides of an equal sign. We want to extend the properties of equality to include taking the square root and taking the cube root of both sides of an equation. . Consider the equality 25 = 25. What happens when we take the square root of both sides of the equal sign? Do we get a true number sentence?  When we take the square root of both sides of the equal sign, we get 5 = 5. Yes, we get a true number sentence. . Consider the equality 27 = 27. What happens when we take the cube root of both sides of the equal sign? Do we get a true number sentence?  When we take the cube root of both sides of the equal sign, we get 3 = 3. Yes, we get a true number sentence. . At this point, we only know that the properties of equality can extend to those numbers that are perfect squares and perfect cubes, but it is enough to allow us to begin solving equations using square and cube roots.

Discussion (5 minutes) The properties of equality have been proven for rational numbers, which are central in school mathematics. As we begin to solve equations that require roots, we are confronted with the fact that we may be working with irrational numbers, which have not yet been defined for students. Therefore, we make the assumption that all of the properties of equality for rational numbers are also true for irrational numbers (i.e., the real numbers, as far as computations are concerned). This is sometimes called the fundamental assumption of school mathematics (FASM). In the discussion below, we reference 푛th roots. You may choose to discuss square and cube roots only. . In the past, we have determined the length of the missing side of a right triangle, 푥, when 푥2 = 25. What is that value, and how did you get the answer?  The value of 푥 is 5 because 푥2 means 푥 ⋅ 푥. Since 5 × 5 = 25, 푥 must be 5. . If we did not know that we were trying to find the length of the side of a triangle, then the answer could also be −5 because −5 × −5 = 25. However, because we were trying to determine the length of the side of a triangle, the answer must be positive because a length of −5 does not make sense. . Now that we know that positive square roots exist and are unique, we can begin solving equations that require roots.

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. When we solve equations that contain roots, we do what we do for all properties of equality (i.e., we apply the operation to both sides of the equal sign). In terms of solving a radical equation, if we assume 푥 is positive, then 푥2 = 25 √푥2 = √25 푥 = √25 푥 = 5. . Explain the first step in solving this equation.  The first step is to take the square root of both sides of the equation. . It is by definition that when we use the symbol √ , it automatically denotes a positive number; therefore, the solution to this equation is 5. In Algebra I, you learn how to solve equations of this form without using the square root symbol, which means the possible values for 푥 can be both 5 and −5 because 52 = 25 and (−5)2 = 25, but for now, we will only look for the positive solution(s) to our equations. 푛 . The symbol √ is called a radical. An equation that contains that symbol is referred to as a radical equation. So far, we have only worked with square roots (i.e., 푛 = 2). Technically, we would denote a square root as 2 √ , but it is understood that the symbol √ alone represents a square root. 3 . When 푛 = 3, then the symbol √ is used to denote the cube root of a number. Since 푥3 = 푥 ∙ 푥 ∙ 푥, the cube root of 푥3 is 푥 (i.e., 3√푥3 = 푥). . For what value of 푥 is the equation 푥3 = 8 true?  푥3 = 8

3 3 √푥3 = √8 3 푥 = √8 푥 = 2

푛 . The 푛th root of a number is denoted by √ . In the context of our learning, we limit our work with radicals to square and cube roots.

Exercises 1–6 (8 minutes) Students complete Exercises 1–6 independently. Allow them to use a calculator to check their answers. Also consider showing students how to use the calculator to find the square root of a number.

Exercises

Find the positive value of 풙 that makes each equation true. Check your solution.

1. 풙ퟐ = ퟏퟔퟗ a. Explain the first step in solving this equation.

The first step is to take the square root of both sides of the equation.

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b. Solve the equation, and check your answer.

풙ퟐ = ퟏퟔퟗ Check:

√풙ퟐ = √ퟏퟔퟗ ퟏퟑퟐ = ퟏퟔퟗ 풙 = √ퟏퟔퟗ ퟏퟔퟗ = ퟏퟔퟗ 풙 = ퟏퟑ

2. A square-shaped park has an area of ퟑퟐퟒ 퐲퐝ퟐ. What are the dimensions of the park? Write and solve an equation.

풙ퟐ = ퟑퟐퟒ Check:

√풙ퟐ = √ퟑퟐퟒ ퟏퟖퟐ = ퟑퟐퟒ 풙 = √ퟑퟐퟒ ퟑퟐퟒ = ퟑퟐퟒ 풙 = ퟏퟖ

The square park is ퟏퟖ 퐲퐝. in length and ퟏퟖ 퐲퐝. in width.

3. ퟔퟐퟓ = 풙ퟐ

ퟔퟐퟓ = 풙ퟐ Check:

√ퟔퟐퟓ = √풙ퟐ ퟔퟐퟓ = ퟐퟓퟐ √ퟔퟐퟓ = 풙 ퟔퟐퟓ = ퟔퟐퟓ ퟐퟓ = 풙

4. A cube has a volume of ퟐퟕ 퐢퐧ퟑ. What is the measure of one of its sides? Write and solve an equation.

ퟐퟕ = 풙ퟑ Check:

ퟑ ퟑ √ퟐퟕ = √풙ퟑ ퟐퟕ = ퟑퟑ ퟑ √ퟐퟕ = 풙 ퟐퟕ = ퟐퟕ ퟑ = 풙

The cube has side lengths of ퟑ 퐢퐧.

5. What positive value of 풙 makes the following equation true: 풙ퟐ = ퟔퟒ? Explain.

풙ퟐ = ퟔퟒ Check:

√풙ퟐ = √ퟔퟒ ퟖퟐ = ퟔퟒ 풙 = √ퟔퟒ ퟔퟒ = ퟔퟒ 풙 = ퟖ

To solve the equation, I need to find the positive value of 풙 so that when it is squared, it is equal to ퟔퟒ. Therefore, I can take the square root of both sides of the equation. The square root of 풙ퟐ, √풙ퟐ, is 풙 because 풙ퟐ = 풙 ⋅ 풙. The square root of ퟔퟒ, √ퟔퟒ, is ퟖ because ퟔퟒ = ퟖ ⋅ ퟖ. Therefore, 풙 = ퟖ.

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6. What positive value of 풙 makes the following equation true: 풙ퟑ = ퟔퟒ? Explain.

풙ퟑ = ퟔퟒ Check:

ퟑ ퟑ √풙ퟑ = √ퟔퟒ ퟒퟑ = ퟔퟒ ퟑ 풙 = √ퟔퟒ ퟔퟒ = ퟔퟒ 풙 = ퟒ

To solve the equation, I need to find the positive value of 풙 so that when it is cubed, it is equal to ퟔퟒ. Therefore, I can take the cube root of both sides of the equation. The cube root of 풙ퟑ, ퟑ√풙ퟑ, is 풙 because 풙ퟑ = 풙 ⋅ 풙 ⋅ 풙. The cube root of ퟔퟒ, ퟑ√ퟔퟒ, is ퟒ because ퟔퟒ = ퟒ ⋅ ퟒ ⋅ ퟒ. Therefore, 풙 = ퟒ.

Discussion (5 minutes) . Now let’s try to identify some roots of some more interesting numbers. Can you determine each of the following?

1 1 3 8 √ √ √ 4 81 27 Provide students a few minutes to work with a partner to find the roots of the above numbers.

1 1 1 1 3 8 2 √ = , √ = , √ = 4 2 81 9 27 3 . Let’s extend this thinking to equations. Consider the equation 푥2 = 25−1. What is another way to write 25−1? 1  The number 25−1 is the same as . 25 . Again, assuming that 푥 is positive, we can solve the equation as before. 푥2 = 25−1 1 푥2 = 25 1 √푥2 = √ 25

1 푥 = √ 25 1 푥 = 5 1 2 1 We know we are correct because ( ) = = 25−1. 5 25

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Exercises 7–9 (4 minutes) Students complete Exercises 7–9 independently. Allow them to use a calculator to check their answers. Also consider showing students how to use the calculator to find the square root of a number.

7. Find the positive value of 풙 that makes the equation true: 풙ퟐ = ퟐퟓퟔ−ퟏ.

풙ퟐ = ퟐퟓퟔ−ퟏ Check:

√풙ퟐ = √ퟐퟓퟔ−ퟏ (ퟏퟔ−ퟏ)ퟐ = ퟐퟓퟔ−ퟏ 풙 = √ퟐퟓퟔ−ퟏ ퟏퟔ−ퟐ = ퟐퟓퟔ−ퟏ ퟏ ퟏ = ퟐퟓퟔ−ퟏ 풙 = √ ퟏퟔퟐ ퟐퟓퟔ ퟏ = ퟐퟓퟔ−ퟏ ퟏ ퟐퟓퟔ 풙 = ퟏퟔ ퟐퟓퟔ−ퟏ = ퟐퟓퟔ−ퟏ 풙 = ퟏퟔ−ퟏ

8. Find the positive value of 풙 that makes the equation true: 풙ퟑ = ퟑퟒퟑ−ퟏ.

풙ퟑ = ퟑퟒퟑ−ퟏ Check:

ퟑ ퟑ √풙ퟑ = √ퟑퟒퟑ−ퟏ (ퟕ−ퟏ)ퟑ = ퟑퟒퟑ−ퟏ 풙 = ퟑ√ퟑퟒퟑ−ퟏ ퟕ−ퟑ = ퟑퟒퟑ−ퟏ ퟏ ퟑ ퟏ = ퟑퟒퟑ−ퟏ 풙 = √ ퟕퟑ ퟑퟒퟑ ퟏ = ퟑퟒퟑ−ퟏ ퟏ ퟑퟒퟑ 풙 = ퟕ ퟑퟒퟑ−ퟏ = ퟑퟒퟑ−ퟏ 풙 = ퟕ−ퟏ

9. Is ퟔ a solution to the equation 풙ퟐ − ퟒ = ퟓ풙? Explain why or why not.

ퟔퟐ − ퟒ = ퟓ(ퟔ) ퟑퟔ − ퟒ = ퟑퟎ MP.6 ퟑퟐ ≠ ퟑퟎ

No, ퟔ is not a solution to the equation 풙ퟐ − ퟒ = ퟓ풙. When the number is substituted into the equation and simplified, the left side of the equation and the right side of the equation are not equal; in other words, it is not a true number sentence. Since the number ퟔ does not satisfy the equation, it is not a solution to the equation.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 8•7

Closing (5 minutes) Summarize, or ask students to summarize, the main points from the lesson: . We know that the positive 푛th root of a number exists and is unique. . We know how to solve equations that contain exponents of 2 and 3; we must use square roots and cube roots.

Lesson Summary

The symbol 풏√ is called a radical. An equation that contains that symbol is referred to as a radical equation. So far, we have only worked with square roots (i.e., 풏 = ퟐ). Technically, we would denote a positive square root as ퟐ√ , but it is understood that the symbol √ alone represents a positive square root.

When 풏 = ퟑ, then the symbol ퟑ√ is used to denote the cube root of a number. Since 풙ퟑ = 풙 ∙ 풙 ∙ 풙, the cube root of 풙ퟑ is 풙 (i.e., ퟑ√풙ퟑ = 풙).

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.

Exit Ticket (5 minutes)

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Name Date

Lesson 3: Existence and Uniqueness of Square Roots and Cube Roots

Exit Ticket

Find the positive value of 푥 that makes each equation true. Check your solution.

1. 푥2 = 225 a. Explain the first step in solving this equation.

b. Solve and check your solution.

2. 푥3 = 64

3. 푥2 = 361−1

4. 푥3 = 1000−1

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Exit Ticket Sample Solutions

Find the positive value of 풙 that makes each equation true. Check your solution.

1. 풙ퟐ = ퟐퟐퟓ a. Explain the first step in solving this equation.

The first step is to take the square root of both sides of the equation.

b. Solve and check your solution.

풙ퟐ = ퟐퟐퟓ Check:

√풙ퟐ = √ퟐퟐퟓ ퟏퟓퟐ = ퟐퟐퟓ 풙 = √ퟐퟐퟓ ퟐퟐퟓ = ퟐퟐퟓ 풙 = ퟏퟓ

2. 풙ퟑ = ퟔퟒ

풙ퟑ = ퟔퟒ Check:

ퟑ ퟑ √풙ퟑ = √ퟔퟒ ퟒퟑ = ퟔퟒ ퟑ 풙 = √ퟔퟒ ퟔퟒ = ퟔퟒ 풙 = ퟒ

3. 풙ퟐ = ퟑퟔퟏ−ퟏ

풙ퟐ = ퟑퟔퟏ−ퟏ Check:

√풙ퟐ = √ퟑퟔퟏ−ퟏ (ퟏퟗ−ퟏ)ퟐ = ퟑퟔퟏ−ퟏ 풙 = √ퟑퟔퟏ−ퟏ ퟏퟗ−ퟐ = ퟑퟔퟏ−ퟏ ퟏ ퟏ = ퟑퟔퟏ−ퟏ 풙 = √ ퟏퟗퟐ ퟑퟔퟏ ퟏ = ퟑퟔퟏ−ퟏ ퟏ ퟑퟔퟏ 풙 = ퟏퟗ ퟑퟔퟏ−ퟏ = ퟑퟔퟏ−ퟏ 풙 = ퟏퟗ−ퟏ

4. 풙ퟑ = ퟏퟎퟎퟎ−ퟏ

풙ퟑ = ퟏퟎퟎퟎ−ퟏ Check:

ퟑ ퟑ √풙ퟑ = √ퟏퟎퟎퟎ−ퟏ (ퟏퟎ−ퟏ)ퟑ = ퟏퟎퟎퟎ−ퟏ 풙 = ퟑ√ퟏퟎퟎퟎ−ퟏ ퟏퟎ−ퟑ = ퟏퟎퟎퟎ−ퟏ ퟏ ퟑ ퟏ = ퟏퟎퟎퟎ−ퟏ 풙 = √ ퟏퟎퟑ ퟏퟎퟎퟎ ퟏ = ퟏퟎퟎퟎ−ퟏ ퟏ ퟏퟎퟎퟎ 풙 = ퟏퟎ ퟏퟎퟎퟎ−ퟏ = ퟏퟎퟎퟎ−ퟏ 풙 = ퟏퟎ−ퟏ

Lesson 3: Existence and Uniqueness of Square Roots and Cube Roots 47

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 8•7

Problem Set Sample Solutions

Find the positive value of 풙 that makes each equation true. Check your solution.

1. What positive value of 풙 makes the following equation true: 풙ퟐ = ퟐퟖퟗ? Explain.

풙ퟐ = ퟐퟖퟗ Check:

√풙ퟐ = √ퟐퟖퟗ ퟏퟕퟐ = ퟐퟖퟗ 풙 = √ퟐퟖퟗ ퟐퟖퟗ = ퟐퟖퟗ 풙 = ퟏퟕ

To solve the equation, I need to find the positive value of 풙 so that when it is squared, it is equal to ퟐퟖퟗ. Therefore, I can take the square root of both sides of the equation. The square root of 풙ퟐ, √풙ퟐ, is 풙 because 풙ퟐ = 풙 ⋅ 풙. The square root of ퟐퟖퟗ, √ퟐퟖퟗ, is ퟏퟕ because ퟐퟖퟗ = ퟏퟕ ⋅ ퟏퟕ. Therefore, 풙 = ퟏퟕ.

2. A square-shaped park has an area of ퟒퟎퟎ 퐲퐝ퟐ. What are the dimensions of the park? Write and solve an equation.

풙ퟐ = ퟒퟎퟎ Check:

√풙ퟐ = √ퟒퟎퟎ ퟐퟎퟐ = ퟒퟎퟎ 풙 = √ퟒퟎퟎ ퟒퟎퟎ = ퟒퟎퟎ 풙 = ퟐퟎ

The square park is ퟐퟎ 퐲퐝. in length and ퟐퟎ 퐲퐝. in width.

3. A cube has a volume of ퟔퟒ 퐢퐧ퟑ. What is the measure of one of its sides? Write and solve an equation.

풙ퟑ = ퟔퟒ Check:

ퟑ ퟑ √풙ퟑ = √ퟔퟒ ퟒퟑ = ퟔퟒ ퟑ 풙 = √ퟔퟒ ퟔퟒ = ퟔퟒ 풙 = ퟒ

The cube has a side length of ퟒ 퐢퐧.

4. What positive value of 풙 makes the following equation true: ퟏퟐퟓ = 풙ퟑ? Explain.

ퟏퟐퟓ = 풙ퟑ Check:

ퟑ ퟑ √ퟏퟐퟓ = √풙ퟑ ퟏퟐퟓ = ퟓퟑ ퟑ √ퟏퟐퟓ = 풙 ퟏퟐퟓ = ퟏퟐퟓ ퟓ = 풙

To solve the equation, I need to find the positive value of 풙 so that when it is cubed, it is equal to ퟏퟐퟓ. Therefore, I can take the cube root of both sides of the equation. The cube root of 풙ퟑ, ퟑ√풙ퟑ, is 풙 because 풙ퟑ = 풙 ⋅ 풙 ⋅ 풙. The cube root of ퟏퟐퟓ, ퟑ√ퟏퟐퟓ, is ퟓ because ퟏퟐퟓ = ퟓ ⋅ ퟓ ⋅ ퟓ. Therefore, 풙 = ퟓ.

5. Find the positive value of 풙 that makes the equation true: 풙ퟐ = ퟒퟒퟏ−ퟏ. a. Explain the first step in solving this equation.

The first step is to take the square root of both sides of the equation.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 8•7

b. Solve and check your solution. Check: 풙ퟐ = ퟒퟒퟏ−ퟏ (ퟐퟏ−ퟏ)ퟐ = ퟒퟒퟏ−ퟏ √풙ퟐ = √ퟒퟒퟏ−ퟏ ퟐퟏ−ퟐ = ퟒퟒퟏ−ퟏ 풙 = √ퟒퟒퟏ−ퟏ ퟏ = ퟒퟒퟏ−ퟏ ퟏ ퟐퟏퟐ 풙 = √ ퟏ ퟒퟒퟏ = ퟒퟒퟏ−ퟏ ퟒퟒퟏ ퟏ 풙 = ퟒퟒퟏ−ퟏ = ퟒퟒퟏ−ퟏ ퟐퟏ 풙 = ퟐퟏ−ퟏ

6. Find the positive value of 풙 that makes the equation true: 풙ퟑ = ퟏퟐퟓ−ퟏ. Check: 풙ퟑ = ퟏퟐퟓ−ퟏ −ퟏ ퟑ −ퟏ ퟑ ퟑ (ퟓ ) = ퟏퟐퟓ √풙ퟑ = √ퟏퟐퟓ−ퟏ ퟓ−ퟑ = ퟏퟐퟓ−ퟏ 풙 = ퟑ√ퟏퟐퟓ−ퟏ ퟏ = ퟏퟐퟓ−ퟏ ퟑ ퟏ ퟓퟑ 풙 = √ ퟏ ퟏퟐퟓ = ퟏퟐퟓ−ퟏ ퟏퟐퟓ ퟏ 풙 = ퟏퟐퟓ−ퟏ = ퟏퟐퟓ−ퟏ ퟓ 풙 = ퟓ−ퟏ

7. The area of a square is ퟏퟗퟔ 퐢퐧ퟐ. What is the length of one side of the square? Write and solve an equation, and then check your solution.

Let 풙 퐢퐧. represent the length of one side of the square.

Check: 풙ퟐ = ퟏퟗퟔ ퟏퟒퟐ = ퟏퟗퟔ √풙ퟐ = √ퟏퟗퟔ ퟏퟗퟔ = ퟏퟗퟔ 풙 = √ퟏퟗퟔ 풙 = ퟏퟒ

The length of one side of the square is ퟏퟒ 퐢퐧.

8. The volume of a cube is ퟕퟐퟗ 퐜퐦ퟑ. What is the length of one side of the cube? Write and solve an equation, and then check your solution.

Let 풙 퐜퐦 represent the length of one side of the cube.

Check: 풙ퟑ = ퟕퟐퟗ ퟑ ퟑ ퟑ ퟗ = ퟕퟐퟗ √풙ퟑ = √ퟕퟐퟗ ퟑ ퟕퟐퟗ = ퟕퟐퟗ 풙 = √ퟕퟐퟗ 풙 = ퟗ

The length of one side of the cube is ퟗ 퐜퐦.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 8•7

9. What positive value of 풙 would make the following equation true: ퟏퟗ + 풙ퟐ = ퟔퟖ?

ퟏퟗ + 풙ퟐ = ퟔퟖ ퟏퟗ − ퟏퟗ + 풙ퟐ = ퟔퟖ − ퟏퟗ 풙ퟐ = ퟒퟗ 풙 = ퟕ

The positive value for 풙 that makes the equation true is ퟕ.

Lesson 3: Existence and Uniqueness of Square Roots and Cube Roots 50