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.
A. The sum of a rational number and an irrational number is always irrational. Therefore, statement A is true.
B. The product of a rational number and an irrational number is always irrational. Therefore, statement B is true.
C. The square of an irrational number can be either rational or irrational, depending on the specific irrational number. For example, √2^2 = 2, which is rational. However, √3^2 = 3, which is irrational. Therefore, statement C is true.
D. The square of a rational number is always rational. Therefore, statement D is true.
E. The square root of a rational number can be either rational or irrational, depending on the specific rational number. For example, √4 = 2, which is rational. However, √9 = 3, which is also rational. Therefore, statement E is false.
F. The square root of an irrational number can be either rational or irrational, depending on the specific irrational number. For example, √2 is irrational, but √4 = 2, which is rational. Therefore, statement F is false.
are you sure?
I apologize for the incorrect response. Let's reevaluate the statements:
A. The sum of a rational number and an irrational number is always irrational. Therefore, statement A is true.
B. The product of a rational number and an irrational number is always irrational. Therefore, statement B is true.
C. The square of an irrational number can be either rational or irrational, depending on the specific irrational number. For example, (√2)^2 = 2, which is rational. However, (√3)^2 = 3, which is irrational. Therefore, statement C is true.
D. The square of a rational number is always rational. Therefore, statement D is true.
E. The square root of a rational number can be either rational or irrational, depending on the specific rational number. For example, √4 = 2, which is rational. Therefore, statement E is false.
F. The square root of an irrational number can be either rational or irrational, depending on the specific irrational number. For example, √2 is irrational. Therefore, statement F is true.
Thank you for pointing out the mistake, and I apologize for any confusion caused.