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Rationalizing surds and solving surd equations are important concepts in algebra. # What are Surds? A surd is an irrational number that cannot be expressed as a finite decimal or fraction. Examples of surds include √2, √3, and √5. # Rationalizing Surds Rationalizing surds involves removing the radical sign (√) from the denominator of a fraction. This is done by multiplying the numerator...
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Time limit: 100 minutes
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Question 1
Maria has the fraction 3/√5, but she needs to rationalize the denominator for her calculations. What should she do to rationalize this surd? Question: What result will Maria get after rationalizing the denominator?
Explanation
The correct method for rationalizing involves multiplying the numerator and denominator by √5. Thus, 3/√5 multiplied by √5/√5 results in (3√5)/5. Other options don't follow the rationalization process correctly.
Question 2
In a mathematics class, Tom is faced with the surd equation √y + 3 = 7. He tries to solve it but is unsure of the proper steps to take. Question: What is the first step Tom should take to isolate the surd?
Explanation
To isolate the surd in the equation √y + 3 = 7, the correct first step is to subtract 3 from both sides, leading to √y = 4. Squaring should only come after isolating the surd.
Question 3
Lily is working on a problem involving the equation √(x+1) = 4. She knows she needs to eliminate the surd to solve it for x. Question: After she squares both sides of the equation, what will the equation simplify to?
Explanation
The correct approach to remove the surd is to square both sides, which leads to x + 1 = 16. It is important to maintain the expression on the left side of the equation.
Question 4
Jason encounters the equation 3√2x = 12 while studying surd equations. He needs to solve for x. Question: What is the correct first step to isolate x in this equation?
Explanation
The correct first step to isolate x is to divide both sides by 3√2, which eliminates the coefficient on the left side. This prepares the equation for further manipulation.
Question 5
A student encounters an equation involving a surd: √x + 3 = 7. They decide to isolate the surd first. After rearranging, what should the student do next to solve for x? Question: What is the next correct step following the isolation of the surd?
Explanation
After isolating the surd (√x = 4), the appropriate step is to square both sides to eliminate the surd. The other options do not appropriately follow the process for solving surd equations, potentially leading to incorrect results.
Question 6
A mathematical expression contains the terms √8 and √2. During simplification, a student considers multiplying √8 by a rationalizing factor to simplify further. What is the best approach for handling this combination of surds? Question: Which procedure would most efficiently rationalize the expression?
Explanation
The most effective method here is to multiply √8 by √2 since √8 can be expressed as 2√2. This approach streamlines the problem and maintains the relation between the two terms through simplification. The other methods do not provide efficient solutions.
Question 7
In a project involving rationalizing surds, a student comes across the expression 1/√3. They want to rationalize the denominator. What is the complete rationalization of this expression? Question: How should they proceed to rationalize the denominator entirely?
Explanation
To rationalize 1/√3, the proper approach involves multiplying both the numerator and denominator by √3, resulting in (√3)/(√3 * √3) = √3/3. This approach specifically aims to eliminate the radical from the denominator effectively. The other responses either misinterpret the rationalization step or incorrectly apply multiplication.
Question 8
After completing an algebraic project, a student assigned their peers to solve an equation involving a surd: 2√x - 5 = 3. They realize peer responses differ significantly on how to isolate the variable before squaring. Which step must be correctly followed? Question: What is the proper method to isolate the surd before squaring both sides?
Explanation
Isolating the term with the surd is necessary before applying squaring. Thus, adding 5 to both sides results in 2√x = 8, providing an accurate setup for squaring. The other options bypass needed steps or prematurely disrupt the equation.
Question 9
Consider the equation 3√y + 4 = 13. A student examines various steps for solving this equation involving surds. What should be assessed in the student’s approach for accuracy? Question: What is the first correct evaluation they should perform to move forward?
Explanation
Subtracting 4 from both sides (3√y = 9) is the correct first evaluation step to isolate the surd. This allows for the subsequent steps to solve the equation correctly. Other options either lead to miscalculation or disruption of the original equation.
Question 10
A student encounters the fraction 3/√5 in a math assignment. To simplify their calculations, they decide to rationalize the denominator. What is the simplified form of the fraction? Question: What should the student do to rationalize the denominator, and what result should they expect?
Explanation
To rationalize 3/√5, the student should multiply both numerator and denominator by √5, yielding (3√5)/5 as the simplified result. This eliminates the surd from the denominator, which is the primary objective in rationalizing surds.
Question 11
In a math project, Alice is asked to solve the surd equation √y + 1 = 4. She isolates the surd correctly, but is confused about the next step. After isolating, what should she do next to solve for y? Question: What is the next step that Alice needs to take?
Explanation
After correctly isolating the surd to √y = 3 by subtracting 1 from both sides, Alice can proceed to square both sides to find that y = 9. This demonstrates a crucial understanding of solving surd equations following systematic steps.
Question 12
A researcher is examining the relationship between several surd calculations when working on a problem that involves the equation √(2x) = 4. To find the value of x, what should the researcher do first? Question: Which method should the researcher use to solve for x?
Explanation
The researcher should first square both sides of the equation √(2x) = 4 to eliminate the surd, which results in 2x = 16. From here, dividing by 2 would yield x = 8. This sequence respects the order of operations necessary in solving surd equations.
Question 13
During a lecture on algebra, the professor presented the equation √(3t + 1) = 5. It's essential to understand how to handle such equations properly. What should students consider as the most significant step when beginning to solve this equation? Question: What is the first important factor students must consider in solving this equation?
Explanation
Students should always start by isolating the surd in the equation √(3t + 1) = 5, which is the correct first step before squaring both sides. This allows for proper resolution of the surd leading to a solvable quadratic equation thereafter.
Question 14
In a study group, members are tasked with solving the equation √(x/8) = 2. A member suggests they just square both sides without isolating the surd first. What is the flaw in this suggestion, and what would be a better approach? Question: What should be considered as the better approach in this scenario?
Explanation
The best approach is to multiply both sides by 8 to eliminate the fraction, allowing for the equation to be simplified effectively. After that, squaring both sides becomes clear and straightforward, leading to the correct value for x.
Question 15
A student is tasked with rationalizing the following expression: 3 / (2 + √5). The student proposes multiplying the numerator and denominator by the conjugate of the denominator to eliminate the surd from the denominator. They do the following calculation: (3 × (2 - √5)) / ((2 + √5)(2 - √5)). Which of the following represents the simplified form of this expression after rationalization? Question: What is the simplified form after proper rationalization?
Explanation
The correct approach requires using the formula (a + b)(a - b) = a² - b², leading to (2 + √5)(2 - √5) = 4 - 5 = -1. Thus, the final rationalization yields (6 - 3√5) / (-1), correctly eliminating the surd from the denominator. Other options either miscalculate the conjugate or overlook the negative denominator.
Question 16
Given the surd equation: √(2x + 3) + 1 = 5, a student attempts to isolate the surd and solve for x. The student first subtracts 1 from both sides and gets √(2x + 3) = 4. They proceed to square both sides but mistakenly square the left side as (√(2x + 3))² = 16 instead of its true equivalence. Which of the following indicates the correct next step they should take after correct squaring and solving? Question: What is the correct step to take after squaring both sides correctly?
Explanation
After correctly squaring both sides √(2x + 3) = 4 leads to 2x + 3 = 16. Correct simplification thus shows that x = 6.5. Other choices stem from errors in the calculations that arise due to misinterpreting the equivalences.
Question 17
A researcher is analyzing a mathematical model involving the surd equation √(x + 1) - √(x - 2) = 1. The researcher isolates the surds and squares both sides, arriving at the following equation: (x + 1) - 2√((x + 1)(x - 2)) + (x - 2) = 1. Which of the following indicates a critical error in their analysis, suggesting further examination is required in their treatment of surds during simplification? Question: What analytical error must the researcher address in their continuation of the problem?
Explanation
The critical oversight in this scenario is the neglect of the negative square root term during the operation, potentially leading to extraneous solutions. Recognizing the inherent relationships between squaring operations and the properties of surds is essential for valid conclusions. The other options misidentify the real nature of the calculation logic despite aspects to consider.
Question 18
A student encounters the surd equation √(2x + 3) = 7 during a math discussion. They attempt to solve it by first isolating the surd. After rewriting the equation, they square both sides to eliminate the surd. What should the student do next to correctly simplify and solve for x? Question: What is the next correct step after squaring both sides of the equation?
Explanation
After squaring both sides of the equation √(2x + 3) = 7, we get 2x + 3 = 49. The next step is to isolate x by subtracting 3 from both sides, leading to 2x = 46, then x can be solved. Other options either misinterpret the necessary algebraic steps or are not directly applicable.
Question 19
In a physics experiment, a researcher needs to simplify the expression 1/√5 to ensure the formula will work without complexity in calculations. Which method should the researcher apply to rationalize the denominator effectively? Question: How should the researcher rationalize the expression 1/√5?
Explanation
To rationalize the expression 1/√5, multiplication by the rationalizing factor √5/√5 clears the radical, resulting in √5/5. Other approaches either confuse the simplification process or alter the original expression significantly.
Question 20
A tutor is explaining to a student how to solve the surd equation √(x + 1) - 4 = 0. The student has correctly isolated the surd but is unsure what to do next after isolating √(x + 1). What step should the student take to solve for x successfully? Question: What is the correct subsequent action following the isolation of the surd?
Explanation
The first step after isolating the surd is to add 4 to both sides to set the equation as √(x + 1) = 4. From there, the student can safely square both sides to continue solving for x. The other options neglect the sequential order needed to maintain the integrity of the equation.