# Tagged Questions

127 views

### Check that $\sqrt{7}(\sqrt[3]{5} - \sqrt[5]{3})$ is not rational

How to prove that $\sqrt{7}(\sqrt[3]{5} - \sqrt[5]{3})$ is not rational. I will appreciate any proof, but I had such exercise during lecture in field theory. Thanks.
103 views

### When to rationalize numerator and/or denominator?

Sometimes, we have to rationalize either the numerator or the denominator, and sometimes we can still work the problem without rationalizing. So, in some cases, rationalizing can be done, although it ...
334 views

### Sum of two irrational radicals is irrational?

If $a,b,m$ and $n$ are positive integers such that $\sqrt[m]{a}$ and $\sqrt[n]{b}$ are irrational numbers, how can we prove that the sum $\sqrt[m]{a}+\sqrt[n]{b}$ is also irrational?
82 views

### $\pi$ does not lie in any quadratic extension of $\mathbb{Q}$

Knowing that $\pi^2$ is irrational: How can we prove that $\pi$ does not lie in any quadratic extension of $\mathbb{Q}$ ? Without using that $\pi$ is transcendent. Any hints would be appreciated.
514 views

### Deciding whether $2^{\sqrt2}$ is irrational/transcendental

Is $2^\sqrt{2}$ irrational? Is it transcendental?
118 views

### Is there a rational univariat polynomial of degree 3 with 3 irrational roots?

The title pretty much asks my question: Does $f\in\mathbb{Q}[x]$ such that $$f(x)=(x-\alpha_1)(x-\alpha_2)(x-\alpha_3),$$ where $\alpha_1, \alpha_2, \alpha_3\in\mathbb{R}\setminus\mathbb{Q}\$ ...
2k views

### How to prove that $\sqrt 3$ is an irrational number? [duplicate]

Possible Duplicate: $a^{1/2}$ is either an integer or an irrational number I know how to prove $\sqrt 2$ is an irrational number. Who can tell me that why $\sqrt 3$ is a an irrational ...
### $\sqrt a$ is either an integer or an irrational number.
I got this interesting question in my mind: How do we prove that if $a \in \mathbb N$, then $\sqrt a$ is an integer or an irrational number? Can we extend this result? That is, can it be shown ...
Is there any theory (analogous to Galois theory) for solving equations with irrational exponents like: $x^{\sqrt{2}}+x^{\sqrt{3}}=1$ ?