SOLUTION: suppose that q is a rational number that is not 0 or 1, and that x is irrational and nonzero. prove that cubic root ((q^2-1)/(qx)) is irrational. please prove by contradiction (

Question 1135667: suppose that q is a rational number that is not 0 or 1, and that x is irrational and nonzero. prove that cubic root ((q^2-1)/(qx)) is irrational.
please prove by contradiction (assuming cubic root ((q^2-1)/(qx)) is rational.)
thanks for helping
Answer by ikleyn(52863) (Show Source): You can put this solution on YOUR website!
.

Let's assume that    is a rational number R:


     = R,, where R is a rational number.


Then  x = , and it is rational number, since the numerator and denominator are rational numbers 

(partly according to the condition and partly according to the assumption).


But we are given that x  is irrational - CONTRADICTION.

The contradiction proves that our assumption is FALSE.

Hence, the number is irrational.

It is exactly what has to be proved.

The proof is completed.

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