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The product is still irrational. Proof by contradiction:
Let X and Y be an element of Q (the set of rational numbers). Thus xy can be written in the form p/q, where p, q are an element of Z (the set of integers), and q is NOT zero. Since X is rational (given in question), it can be written as r/s. Again, r,s are elements of Z and s is not equal to zero.
This gives Y=ps/qr, which is also an element of Q (rationals). However, we know that Y is not rational (given in the question), hence this is the contradiction. Hence the product of a rational and an irrational is irrational.
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How you prove that -if r is rational and x is irrational then prove that r x and rx are rational?
To prove that if (r) is rational and (x) is irrational, then both (rx) and (\frac{r}{x}) are rational, we can use the fact that the product or quotient of a rational and an irrational number is always irrational. Since (r) is rational and (x) is irrational, their product (rx) must be irrational. Similarly, the quotient (\frac{r}{x}) must also be irrational. Therefore, we cannot prove that both (rx) and (\frac{r}{x}) are rational based on the given information.
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Rational
Is 4.46466... rational or irrational?
No
Is it true product of a non zero rational number and irrational number is rational number?
It is always FALSE.
Is -4x rational or irrational?
If x is rational the it is rational. If x is irrational then it is irrational.
