# Tagged Questions

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### Help With a proof (Irrational Number)

Prove the following statement by proving its contrapositive: if $r$ is irrational, then $r^\frac{1}{5}$ is irrational. Its contrapositive will be: If $r^\frac{1}{5}$ is not irrational, then $r$ is ...
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### Help proving a theorem in my textbook

If $r \in \mathbb{N}$ is not a perfect square, then $\sqrt{r}$ is irrational. For reference, an integer $n$ is a perfect square if $n=m^2$ for some $m \in \mathbb{Z}$. Any help proving this ...
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### Help with a proof my professor gave my class

Let $x,y \in \mathbb{R}$ with $x<y$. There exists an irrational number $z$ such that $x<z<y$. My proof so far: Let $x,y \in \mathbb{R}$ and assume $x<y$. Then, by Theorem 11.8 (in ...
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### Prove that if $n$ is a positive integer then $\sqrt{n}+ \sqrt{2}$ is irrational.

The sum of a rational and irrational number is always irrational, that much I know - thus, if n is a perfect square, we are finished. However, is it not possible that the sum of two irrational numbers ...
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### Proving either $x^2$ or $x^3$ is irrational if $x$ is irrational

I had a test today in discrete mathematics and I am dubious whether or not my proof is correct. Suppose $x$ is an irrational number. Prove that either $x^2$ or $x^3$ is irrational. My Answer: ...
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### Prove that there is no rational number solution for an equation.

Prove that there is no rational number solution to the equation $x^2-3x+1=0$. (Note, we do not assume that we know all the solutions of $x^2-3x+1=0$ are given by quadratic formula)