Surds
Question on surds
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Time limit: 50 minutes
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Question 1
What is a surd?
Explanation
A surd is defined as an irrational number that cannot be simplified to remove a root, such as √2.
Question 2
Which of the following is an example of a surd?
Explanation
√3 is an example of a surd because it is an irrational number that cannot be simplified to a whole number.
Question 3
Which expression cannot be classified as a surd?
Explanation
√9 equals 3, which is a rational number. Surds are those that do not simplify to a whole number.
Question 4
What is true about the sum of two surds?
Explanation
The sum of two surds may still be irrational (a surd) or may result in a rational number, depending on the values.
Question 5
Which operation is likely to result in a surd?
Explanation
Taking the square root of a prime number results in a surd because prime numbers do not have exact square roots that are rational.
Question 6
If √50 is expressed in its simplest form, which of the following is the correct expression?
Explanation
The simplest form of √50 is obtained by factoring out perfect squares. √50 = √(25*2) = √25 * √2 = 5√2. The other options present common misconceptions such as miscalculating the factorization or incorrectly combining separate surd terms.
Question 7
When adding 3√2 and 5√2, what is the resulting expression?
Explanation
Adding surds with the same radical expression involves combining the coefficients: 3√2 + 5√2 = (3+5)√2 = 8√2. The distractors involve either incorrect coefficient addition or misrepresentation of radical simplifications, leading to confusion.
Question 8
[Case Scenario] A student in a mathematics class is working through various problems involving surds. They encounter the expression √50 and are attempting to simplify it. After some calculations, the student realizes that they need to break down 50 into its prime factors. They correctly identify that 50 can be expressed as 25 × 2. Question: What is the simplest form of √50 after the student applies their understanding of surds?
Explanation
The expression √50 simplifies to 5√2 because 50 can be broken down into 25 (a perfect square) and 2, allowing for the simplification through the square root properties.
Question 9
[Case Scenario] A high school teacher is explaining how to add and subtract surds to their students. To illustrate the concept, the teacher writes down the expressions 3√2 and 4√2 on the board and asks the students to find the result of adding these two surds together. Question: What is the correct result of adding 3√2 and 4√2?
Explanation
When adding surds, you combine the coefficients of like terms. Here, 3√2 + 4√2 results in 7√2.
Question 10
[Case Scenario] A student is assigned a project to evaluate different properties of surds. In their research, they come across the surd √18 and decides to determine its rational approximation by simplifying it and justifying its usage in different mathematical scenarios. Question: What value does √18 approximate to when simplified and rounded to two decimal places?
Explanation
The simplification of √18 leads to 3√2, which when calculated equals around 4.24. Understanding how to manipulate and approximate surds is crucial for mathematical evaluations.