This lesson covers basic square and cube root equations, such as \(x^2\) = 10 and \(x^3\) = -8. Students will:
- solve simple square root equations, listing both solutions as necessary.
- solve simple cubic root equations.
- determine whether equations of both types have whole number solutions, irrational solutions, or no actual solutions.

- How can mathematics help to quantify, compare, depict, and model numbers?

- Cube root: One of three equal factors of a number.
- Perfect cube: A number whose cube root is a whole number, or the result of cubing a whole number.
- Perfect square: A number whose square root is a whole number, or the result of squaring a whole number.
- Square root: One of two equal factors of a number.

- Square Root Equations worksheet (M-8-4-3_Square Root Equations and KEY) for each student. 
- Cube Root Equations worksheet (M-8-4-3_Cube Root Equations and KEY) for each student. 
- Lesson 3 Exit Ticket (M-8-4-3_Lesson 3 Exit Ticket and KEY) for each student. 
- Index cards for flash cards

- The work that students do with the square root equations can be used to assess them. 
- Students' competency can be measured using the Cube Root Equations worksheet. 
- Use the Lesson 3 Exit Ticket to assess student understanding of concepts. 
 

Scaffolding, Active Engagement, Modeling, and Formative Assessment 
W: Students will learn to solve simple quadratic and cubic equations, such as \(x^2\) = n or \(x^3\) = n, with n as a real number. Students will learn to look at simple quadratic or cubic equations and determine how many real solutions exist. 
H: Students are given a simple problem with a clear positive solution, but many may overlook the negative option. Students will be captivated by the idea that there is another, elusive answer, as well as the ease with which such equations can be solved. 
E: During the teacher-led lesson, students will be given the resources they need to solve basic quadratic and cubic equations. They will also have the chance to learn more about these equations through two worksheets and the Exit Ticket. 
R: Students will update their thinking by completing two worksheets with various problems. Their final opportunity to modify their thinking and synthesize the principles offered is to complete the Exit Ticket, which deals with solving equations in a more general manner. 
E: Check students' comprehension levels after completing worksheets in Activities 1 and 2. 
T: Use the Extension section to customize the lesson to match the needs of the students. The Routine area offers ideas and chances to review course concepts throughout the year. The Small Group portion is designed for students who could benefit from more practice or teaching. The Expansion section contains suggestions for challenges that go beyond the criteria of the standard. 
O: The class begins with a clear problem that engages students and goes swiftly. Students may be amazed at how rapidly they can answer seemingly complex problems, especially as they get into cube roots. Students are then given a worksheet including multiple problems that most students should be able to finish quickly, boosting their confidence. 

Activity 1

On the board, write the equation \(x^2\) = 25. Ask students to find all of the solutions and raise their hands when they are done. When the majority of the class has finished, request all solutions. If they do not provide x = −5, inform them that they are missing one solution.

Demonstrate the fact that it is easy to forget about the negative solution to an equation like \(x^2\) = 25.

"How did you figure out that the solution was x = 5?" Responses are likely to include that 5 × 5 = 25. "How did you determine that x = −5 was a solution?" Use this opportunity to reinforce the principles for multiplying negative numbers, which state that the product of two negative numbers is a positive number. Help students recognize that the square root of 25 is ±5. (Depending on the class, determine whether to introduce the ± symbol.)

"What are all the answers to \(x^2\) = 100?" (±10) "How did you know that x = 10 was a solution?" Encourage students to remember that the square root of 100 is ±10, not just 10 × 10 = 100. The idea is to start students thinking about taking the square root of both sides.

"Keep in mind that, in most cases, we want to obtain the variable—in this case, x—all by itself when solving equations. What operation would we carry out on both sides of the equation to isolate x if the equation was x − 4 = 20?" (Add 4.) "“This works because addition and subtraction are inverse operations. What would happen if 5x = 80 was the equation? " (Divide by 5 ) "Once more, the reason this works is that division and multiplication are inverse operations. To solve our algebraic problem, \(x^2\) = 100, we must determine what the inverse action of squaring something is. Any predictions?" (square root)

"Squaring something and taking the square root are inverse actions, therefore, when we take the square root of both sides, all that's left on the left side is x. They cancel each another out. The square root of 100, which is 10 and −10, is shown on the right-hand side."

Write on the board, \(x^2\) = 31. "What are this equation's solutions?" Students may find this equation difficult to solve or may even say there are no solution. Remind students that isolating the x is their job. Otherwise, eliminate the exponent. Assist students in realizing that x = ± the square root of 31 is the result of taking the square root of both sides. Make sure the negative square root is included for the students. The ± symbol should be introduced now, if it hasn't already, as a useful way to denote both positive and negative solutions.

"Does 31 have a square root?" Students can easily claim that it doesn't in this situation. Mention that although the square root is there, it is not a full number. (If the class has been exposed to rational and irrational numbers, you may choose to mention that it is an irrational number.) "However, it's a decimal that goes on indefinitely with no pattern, so we'll just leave it at the square root of 31, allowing us to avoid rounding our answer. We are certain that our response is accurate."

Verbally give students several additional equations to solve in the form \(x^2\) = p, where p can be either a perfect square (sometimes) or a non-perfect square (comprising fractions, decimals, etc.). When the class appears to be proficient in resolving these kinds of problems, write the subsequent equation on the board:

3\(x^2\) = 90

"Remember, we're looking for x on its own. However, this presents two challenges: first, we must eliminate 3; and second, we must eliminate the exponent. What ought to be eliminated initially?" Permit some debate, pointing out that we must find the square root of 3 if we take the square root of both sides of the equation. Proceed with the following after some discussion: "Consider the sequence of events. We are aware that exponents and roots come before division when we are simplifying an expression. But in this case, we need to utilize the opposite order of operations because we are "undoing" these steps to isolate the variable and solve the problem. This means that we have to take the square root of each side of the equation after dividing both sides by 3 ."

“So, we’ll divide by 3 to get \(x^2\) = 30. What are the solutions to our equation? (±\(\sqrt{30}\))

"Consider a formula like \(x^2\) = -6. What equation's solutions are there?" Let's explore the concept of taking the square root of a negative number and have a little conversation. Inform students that while a negative number can be squared, the result is what is known as an imaginary number. "At this point, we don't need to learn much more about imaginary numbers. However, since there is no real number that can be the square root of -6, we can state that our equation does not have any real solutions."

Give a copy of the Square Root Equations worksheet (M-8-4-3_Square Root Equations and KEY) to every student. Let them finish it for a while before checking their responses and gathering them.

Activity 2

Ask students to solve the equation \(x^3\) = 64 by writing it on the board and guiding them through each step. Responses will vary because students might glance at this problem and think it's a square root problem, which would result in ±8 answers. While some students might observe that it's a cube root, they might believe that the answers are ±4 instead of just 4.

In the event where students provided ±8, emphasize that the variable is being cubed rather than squared.

"How do we know that the equation's solution is 4?" (64 has a cube root of 4, and 4 × 4 × 4 = 64.)

“What about −4? What is the result of −4 × −4 × −4?” (The value is −64. If needed, spend a moment explaining how the sign of \((-4)^3\) becomes negative instead of positive.

"Therefore, is -4 a solution?" (No.) "The only actual solution we have for this cube root problem is 4. That being said, there is one difference between using cube roots and square roots to solve problems. Usually, you have to remember to include the plus or minus sign while working with square roots. Cube roots do not require its inclusion."

“How about \(x^3\) = 8?” (x = 2).

“How about \(x^3\) = −8?" It's possible for students to reply, "There's no solution." Remind them that we produced a negative result when we performed −4 × −4 × −4. This suggests that there may be negative solutions to problems involving \(x^3\). Assist students in understanding that x = -1/2 is the answer.

On the board, write \(x^3\) = 7. "What is this equation's solution?" Students might argue that there isn't one because 7 isn't a perfect cube, just like before. "Remember that isolating the x is our goal. We took the square root when it was equal to x squared. Now that it's x cubed , what should we do?" Even if they've never heard of it, students should be able to estimate the cube root of both sides by paying attention to the repeated term "square."

"Because they are inverse operations, the root, and the exponent cancel each other out when we take the cube root of \(x^3\). All that is left on the left-hand side is an x. The cube root of 7 is what's left on the right." Write x = 3\(\sqrt{7}\) on the board.

"Observe how similar the cube root and square root symbols are to one another. Although our symbol is radical, there is one small but significant difference. To indicate that the radical is a cube root—the opposite of raising anything to the third power—we place a little 3 on its "shelf." Even though we might, since square roots are so frequently found, we don't write a small 2 when we take a square root. This indicates that a square root (or \(\sqrt[2]{x}\)) is present if the radical sign is present without a small number.

"In this case, the cube root of 7 is the answer. Once more, we don't utilize a ± sign here, as cubing a negative integer doesn't yield a positive result."

Write \(x^3\) = -4. "What is the solution for x?" In this case, students must integrate the two new ideas regarding cube roots: that a solution must include a cube root rather than a square root, and that a solution can be negative. students should understand that (x = \(\sqrt[3]{-4}\)) is the answer in this case.

Distribute the Cube Root Equations worksheet (M-8-4-3_Cube Root Equations and KEY) to the students, and allow them time to finish it before collecting and examining their answers.

Students must also finish the Lesson 3 Exit Ticket (M-8-4-3_Lesson 3 Exit Ticket and KEY) before they leave.

Extension:

This lesson can be modified using the following strategies to accommodate students' needs all year long:

Routine: As students gain proficiency in solving increasingly complex equations involving square and cube roots, such as 4\(x^3\) −12 = \(x^3\) + 8, the concepts covered in this lesson can be reinforced throughout the year.

This lesson's concepts can also be given in a geometric context while discussing area and volume. As an illustration, what is the length of one side of a cube whose volume is 125 \(in^3\)? It is necessary to set up and solve the equation \(x^3\) = 125 for questions of that kind.

Small Group: Students can make flash cards and test each other on them to improve their knowledge of common squares and cubes (squares up to 144 and cubes up to 125). In groups, students can also examine how many squares (or cubes) they can learn collectively and compare that number to that of other groups.

Expansion: Students who want a challenge that goes above and beyond what is required by the standard can investigate using their calculators' \(\sqrt[x]{}\) button to solve higher-degree equations like \(x^4\) = 1,024. (While it can also aid students in understanding the sharp rise in power sizes as exponents increase, solving such problems without the \(\sqrt[x]{}\) button may prove frustrating for them.)

Similarly, students might start investigating fractional and negative exponents by using a calculator. Students can assess the impact of the negative sign on the exponent's evaluation by evaluating, for instance, \(n^{-1}\), where n is any integer between 1 and 10.