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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.


Prove R is rational and S is irrational then R plus S must be irrational?

Let R + S = T, and suppose that T is a rational number.The set of rational number is a group.This implies that since R is rational, -R is rational [invertibility].Then, since T and -R are rational, T - R must be rational [closure].But T - R = S which implies that S is rational.That contradicts the fact that y is an irrational number. The contradiction implies that the assumption [that T is rational] is incorrect.Thus, the sum of a rational number R and an irrational number S cannot be rational.


Why the area of a circle with a rational radius must be an irrational number?

It the radius is r then the area is pi*r*r - which is pi times a rational number. pi is an irrational number, so the multiple of pi and a rational number is irrational.


Is 4.6 irrational or rational?

4.6 is rational.


Is 37.5 a rational or irrational number?

Rational


Is 10.01 a rational or irrational number?

10.01 is a rational number


Is 34.5 rational or irrational?

is 34.54 and irrational or rational. number


Is 0.34343434 rational or irrational?

Rational


Is 2.717711777 rational or irrational?

Rational.


How do you write an irrational number in algebra?

There is no representation for irrational numbers: they are represented as real numbers that are not rational. The set of real numbers is R and set of rational numbers is Q so that the set of irrational numbers is the complement if Q in R.


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.