Linked Questions

29
votes
6answers
6k views

How to prove that $\sqrt[3] 2 + \sqrt[3] 4$ is irrational? [duplicate]

So while doing all sorts of proving and disproving statements regarding irrational numbers, I ran into this one and it quite stumped me: Prove that $\sqrt[3]{2} + \sqrt[3]{4}$ is irrational. I tried ...
33
votes
5answers
5k views

Can $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m$ are perfect squares?

Can the expression $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m \in \mathbb{N}$ are perfect squares? It doesn't seem likely, the only way that could happen is if for example $\sqrt{m} = a-\sqrt{n}...
19
votes
5answers
3k views

Prove that $5^{1/3}+7^{1/2}$ is irrational

Goal: Prove that $5^{1/3}+7^{1/2}$ is irrational. Idea: We can prove this is irrational by supposing it is rational and finding a contradiction. So, $5^{1/3}+7^{1/2} = p/q$ where $p$ and $q$ are ...
14
votes
4answers
919 views

Process to show that $\sqrt 2+\sqrt[3] 3$ is irrational

How can I prove that the sum $\sqrt 2+\sqrt[3] 3$ is an irrational number ??
9
votes
3answers
642 views

$\sqrt[31]{12} +\sqrt[12]{31}$ is irrational

Prove that $\sqrt[31]{12} +\sqrt[12]{31}$ is irrational. I would assume that $\sqrt[31]{12} +\sqrt[12]{31}$ is rational and try to find a contradiction. However, I don't know where to start. Can ...
19
votes
2answers
615 views

Sum of irrational numbers, a basic algebra problem

Let $x_1,\dots,x_n$ be positive rational numbers. If $\sqrt[l_1]{x_1},\dots,\sqrt[l_n]{x_n}$ are all irrational numbers (where $l_1,l_2,\dotsc,l_n\in\Bbb N^*$), does it follow that $$\sqrt[l_1]{x_1}+ ...
13
votes
2answers
355 views

If $\sqrt[3]{a} + \sqrt[3]{b}$ is rational then prove $\sqrt[3]{a}$ and $\sqrt[3]{b}$ are rational

Assume there exist some rationals $a, b$ such that $\sqrt[3]{a}, \sqrt[3]{b}$ are irrationals, but: $$\sqrt[3]{a} + \sqrt[3]{b} = \frac{m}{n}$$ for some integers $m, n$ $$\implies \left(\sqrt[3]{a} ...
9
votes
2answers
2k 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?
2
votes
4answers
264 views

Is $a\sqrt[3]{2} + b\sqrt[3]{4}$ irrational?

I need to prove that $$ a\sqrt[3]{2} + b\sqrt[3]{4}$$ is irrational, while $a$,$b $ are non zero rationals. I know that $\sqrt[3]{2} + \sqrt[3]{4}$ is irrational and I also know how to prove it, ...
7
votes
1answer
559 views

Show the two fields are not isomorphic

Let $p,q,r$ be prime integers with $q\neq r$. Let $\sqrt[p]q$ denote any root of $x^p-q$ and $\sqrt[p]r$ denote any root of $x^p-r$. Please show that $\mathbb{Q}(\sqrt[p]q)\neq\mathbb{Q}(\sqrt[p]r)$. ...
0
votes
0answers
108 views

How to prove $a_{1}\sqrt[b_{1}]{c_{1}}+a_{2}\sqrt[b_{2}]{c_{2}}+…+a_{n}\sqrt[b_{n}]{c_{n}}$ is irrational?

Let's define the number $$A=a_{1}\sqrt[b_{1}]{c_{1}}+a_{2}\sqrt[b_{2}]{c_{2}}+.....+a_{n}\sqrt[b_{n}]{c_{n}}$$ where $a_{1}, a_{2}, ..., a_{n}$ are positive integers and $b_{1}, b_{2}, ..., b_{n}, ...