Q: If P = {x : x is a positive perfect square and x < 20}, what is its roster form?

P = {1, 4, 9, 16} Positive perfect squares below 20 are obtained from 1², 2², 3², and 4²: 1, 4, 9, and 16. The next square is 5² = 25, which is not less than 20. Thus the complete roster is {1, 4, 9, 16}. Option B lists even numbers, C lists initial natural numbers, and D includes 25, so A is correct.

Q: If K={1,4,9,16,25}, what is the most suitable description of K?

Perfect squares less than 30 The members of K are successive squares: 1=1², 4=2², 9=3², 16=4², and 25=5². Every one is less than 30, while the next square, 6²=36, is not less than 30. Hence K is precisely the set of perfect squares below 30. The numbers are not all prime or even, and most are not multiples of 5, so option A is correct.

Q: If V={x:x∈N, x is a perfect square less than 36 and x≠4}, what is V?

V={1,9,16,25} The governing concept is filtering a set using two conditions. The natural perfect squares below 36 are 1²=1, 2²=4, 3²=9, 4²=16, and 5²=25. The number 36 is excluded because the condition says less than 36, and 4 is removed by x≠4. Thus V={1,9,16,25}, which is option A.

Q: If C={x:x²=16, x∈Z}, how many elements does C have?

2 The governing concept is solving a square equation over the integers and then counting distinct set elements. From x²=16, we obtain x=4 or x=-4 because both values have square 16. Thus C={-4,4}, which contains two distinct elements. The value 16 is the square on the right side, not the cardinality, and 4 is only one solution. Therefore option B is correct.

Q: Which expression has $25a^2$ as its square?

$5a$ Since $(5a)^2=5^2\times a^2=25a^2$, the correct expression is $5a$. The square of $25a$ is $625a^2$, while the square of $5a^2$ is $25a^4$. Exam tip: to find the square root of a perfect-square monomial, take the square root of the coefficient and halve the exponent of each variable.

Q: Which option is correct?

√4 + √9 = 5 The governing concept is evaluation of principal square roots before carrying out addition. Since 4 = 2², √4 = 2, and since 9 = 3², √9 = 3. Therefore, √4 + √9 = 2 + 3 = 5, so option B is correct. A common error is to combine the radicands and write √(4 + 9) = √13; however, the identity √a + √b = √(a+b) is not generally valid. Thus option A applies an incorrect rule. Option C adds 4 and 9 but does not take their roots, so it does not represent the given expression. Option D has no valid calculation supporting it. The answer is especially clear because both radicands are perfect squares and their roots are whole numbers.

Question 1

How can G = {2, 3, 5, 7} be described most accurately?

Accepted Answer

Prime natural numbers less than 10 A set-builder description must include every listed element and exclude numbers not in the set. The numbers 2, 3, 5, and 7 are all prime natural numbers less than 10, so option A is exact. Option B would contain only 2, 4, 6, and 8. Option C is too broad because it also includes 1 and 9, while option D gives 1, 4, and 9.

Question 2

If P = {x : x is a positive perfect square and x < 20}, what is its roster form?

Accepted Answer

P = {1, 4, 9, 16} Positive perfect squares below 20 are obtained from 1², 2², 3², and 4²: 1, 4, 9, and 16. The next square is 5² = 25, which is not less than 20. Thus the complete roster is {1, 4, 9, 16}. Option B lists even numbers, C lists initial natural numbers, and D includes 25, so A is correct.

Question 3

If K={1,4,9,16,25}, what is the most suitable description of K?

Accepted Answer

Perfect squares less than 30 The members of K are successive squares: 1=1², 4=2², 9=3², 16=4², and 25=5². Every one is less than 30, while the next square, 6²=36, is not less than 30. Hence K is precisely the set of perfect squares below 30. The numbers are not all prime or even, and most are not multiples of 5, so option A is correct.

Question 4

If V={x:x∈N, x is a perfect square less than 36 and x≠4}, what is V?

Accepted Answer

V={1,9,16,25} The governing concept is filtering a set using two conditions. The natural perfect squares below 36 are 1²=1, 2²=4, 3²=9, 4²=16, and 5²=25. The number 36 is excluded because the condition says less than 36, and 4 is removed by x≠4. Thus V={1,9,16,25}, which is option A.

Question 5

If C={x:x²=16, x∈Z}, how many elements does C have?

Accepted Answer

2 The governing concept is solving a square equation over the integers and then counting distinct set elements. From x²=16, we obtain x=4 or x=-4 because both values have square 16. Thus C={-4,4}, which contains two distinct elements. The value 16 is the square on the right side, not the cardinality, and 4 is only one solution. Therefore option B is correct.

Question 6

Which expression has $25a^2$ as its square?

Accepted Answer

$5a$ Since $(5a)^2=5^2\times a^2=25a^2$, the correct expression is $5a$. The square of $25a$ is $625a^2$, while the square of $5a^2$ is $25a^4$. Exam tip: to find the square root of a perfect-square monomial, take the square root of the coefficient and halve the exponent of each variable.

Question 7

Which option is correct?

Accepted Answer

√4 + √9 = 5 The governing concept is evaluation of principal square roots before carrying out addition. Since 4 = 2², √4 = 2, and since 9 = 3², √9 = 3. Therefore, √4 + √9 = 2 + 3 = 5, so option B is correct. A common error is to combine the radicands and write √(4 + 9) = √13; however, the identity √a + √b = √(a+b) is not generally valid. Thus option A applies an incorrect rule. Option C adds 4 and 9 but does not take their roots, so it does not represent the given expression. Option D has no valid calculation supporting it. The answer is especially clear because both radicands are perfect squares and their roots are whole numbers.

Squares and Square Roots

Practice questions

01 How can G = {2, 3, 5, 7} be described most accurately?

02 If P = {x : x is a positive perfect square and x < 20}, what is its roster form?

03 If K={1,4,9,16,25}, what is the most suitable description of K?

04 If V={x:x∈N, x is a perfect square less than 36 and x≠4}, what is V?

05 If C={x:x²=16, x∈Z}, how many elements does C have?

06 Which expression has \(25a^2\) as its square?

07 Which option is correct?

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