let a be a rational number and b be an irrational number. Which of the following statements are true?
A. The sum of a and b is never rational.**
B. The product of a and b is rational.
C. b^2 is sometimes rational.**
D. a^2 is always rational.**
E. √a is never rational.
F. √b is never rational.**
The correct statements are:
A. The sum of a and b is never rational.
C. b^2 is sometimes rational.
D. a^2 is always rational.
F. √b is never rational.
Explanation:
A. The sum of a rational number and an irrational number is always irrational, since the sum would involve a finite decimal part (rational) and an infinite decimal part (irrational). Therefore, the sum of a and b is never rational.
B. The product of a rational number and an irrational number is always irrational. Therefore, the statement "the product of a and b is rational" is false.
C. The square of an irrational number can sometimes be rational. For example, if b = √2, then b^2 = 2, which is rational. Therefore, b^2 is sometimes rational.
D. The square of any rational number is always rational. Therefore, a^2 is always rational.
E. The square root of a rational number can sometimes be rational. For example, if a = 4, then √a = 2, which is rational. Therefore, the statement "√a is never rational" is false.
F. The square root of an irrational number is always irrational. Therefore, √b is never rational.