Question

let a and b be rational numbers

the difference of a and b is

the product of a and b is

\sqrt{ab} is

Answer 1

are they

always rational

sometimes rational

never rational

Answer 2

To clarify, let's define a and b as follows:
a = p/q, where p and q are integers and q is not equal to zero.
b = r/s, where r and s are integers and s is not equal to zero.

The difference of a and b is given by (a - b) which is (p/q) - (r/s):
(a - b) = (ps - rq)/(qs)

The product of a and b is given by (a * b) which is (p/q) * (r/s):
(a * b) = (pr)/(qs)

The square root of the product of a and b, √(ab), is:
√(ab) = √((pr)/(qs)) = √(pr)/√(qs) (where pr and qs are integers and pr is not equal to zero)

Answer 3

In general, the difference of two rational numbers may or may not be rational. It depends on the specific values of a and b. If (ps - rq)/(qs) can be expressed as a fraction of two integers, then the difference is rational. However, if it cannot be expressed as a fraction, then the difference would be irrational.

Similarly, the product of two rational numbers may or may not be rational. If (pr)/(qs) can be expressed as a fraction of two integers, then the product is rational. However, if it cannot be expressed as a fraction, then the product would be irrational.

As for √(ab), it can be either rational or irrational, depending on the specific values of a and b. If the product ab is a perfect square (i.e., the square of an integer), then √(ab) is rational. However, if ab is not a perfect square, then √(ab) would be irrational.

Therefore, the answers to your questions are as follows:

- The difference of a and b can be sometimes rational, sometimes irrational.
- The product of a and b can be sometimes rational, sometimes irrational.
- √(ab) can be sometimes rational, sometimes irrational.