Question

Let a be a rational number and b be an irrational number. Which of the following are true statements?(there is more than 1 answer)
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.) square root of a is never rational

F.) square root of b is never rational

Answer 1

A.) the sum of a and b is never rational.
This is a true statement. Since an irrational umber has a decimal part that is infinite and non-periodical, when you add a rational number to an irrational number, the result will have the same infinite non periodical decimal part, so the new number will be irrational as well.

B.) The product of a and b is rational
This one is false. Zero is a rational number, and when you multiply an irrational number by zero, the result is always zero.

C.) b^2 is sometimes rational
This one is true. When you square an irrational number that comes from a square root like $\sqrt{2}$, you will end with a rational number: $( \sqrt{2} )^{2}=2$, but, if you square rationals from different roots than square root like $\sqrt[3]{2}$, you will end with an irrational number: $\sqrt[3]{2^{2} } = \sqrt[3]{2}$. 

D.) a^2 is always rational
This one is false. If you square a rational number, you will always end with another rational number.

E.) square root of a is never rational
This one is false. The square root of perfect squares are always rational numbers: $\sqrt{64} =8$, $\sqrt{16} =4$,...

F.) square root of b is never rational
This one is true. Since the square root of any non-perfect square number is irrational, and all the irrational numbers are non-perfect squares, the square root of an irrational number is always irrational.

We can conclude that given that a is a rational number and b be an irrational number, A, C, D, and F are true statements.