Squares and Square Roots

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.

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.

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.

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.

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.

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.