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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.
Is it true product of a non zero rational number and irrational number is rational number?
It is always FALSE.
Is 0.34343434 rational or irrational?
Rational
Is 4.46466... rational or irrational?
No
Is -4x rational or irrational?
If x is rational the it is rational. If x is irrational then it is irrational.
What product is true about the irrational and rational numbers?
The product of 2 rationals must be rational. The product of a rational and an irrational is irrational (unless the rational is 0) The product of two irrationals can be either rational or irrational.
What is the product of rational and irrational number?
The product of a rational and irrational number can be rational if the rational is 0. Otherwise it is always irrational.
Is the product of a rational number and an irrational number rational or irrational?
Such a product is always irrational - unless the rational number happens to be zero.
Is the product of a rational number and an irrational number always irrational?
No. 0 is a rational number and the product of 0 and any irrational number will be 0, a rational. Otherwise, though, the product will always be irrational.
What does a rational number times an irrational number equal?
The product of 0 and an irrational is 0 (a rational), the product of a non-zero rational and any irrational is always irrational.
Why is the product of a rational number and an irrational number irrational?
The question cannot be answered because it is based on a false premise.The product of a (not an!) rational number and an irrational number need not be irrational. For eample, the product ofthe rational number, 0, and the irrational number, pi, is 0. The product is rational, not irrational!
What is the product of one irrational number and one rational number?
Provided that the rational number is not 0, the product is irrational.
Is the product of a nonzero rational number and an irrational number rational or irrational?
It is always irrational.
Is the product of a rational number and irrational number always rational?
No.A rational times an irrational is never rational. It is always irrational.
What is the classification of the product of a rational and an irrational number?
If you multiply a rational and an irrational number, the result will be irrational.
Is the product of an irrational number and a rational number always an irrational number?
Not if the rational number is zero. In all other cases, the product is irrational.
Why the product of nonzero rational number and a rational number is an irrational?
Actually the product of a nonzero rational number and another rational number will always be rational.The product of a nonzero rational number and an IRrational number will always be irrational. (You have to include the "nonzero" caveat because zero times an irrational number is zero, which is rational)
