66k views

### Prove that $\sqrt 2 + \sqrt 3$ is irrational [duplicate]

I have proved in earlier exercises of this book that $\sqrt 2$ and $\sqrt 3$ are irrational. Then, the sum of two irrational numbers is an irrational number. Thus, $\sqrt 2 + \sqrt 3$ is irrational. ...
7k views

### Prove that $\sqrt 2 +\sqrt 3$ is irrational. [duplicate]

Please prove that $\sqrt 2 + \sqrt 3$ is irrational. One of the proofs I've seen goes: If $\sqrt 2 +\sqrt 3$ is rational, then consider $(\sqrt 3 +\sqrt 2)(\sqrt 3 -\sqrt 2)=1$, which implies that ...
18k views

### Prove $\sqrt{2} + \sqrt{5}$ is irrational [duplicate]

How do you prove that $\sqrt{2} + \sqrt{5}$ is irrational? I tried to prove it by contradiction and got this equation: $a^2/b^2 = \sqrt{40}$.
2k views

### Proving/disproving that √7 - √2 is irrational [duplicate]

It's been proven that √7 and √2 are irrational. However, I am not sure how to go about proving that √7 - √2. Is it an acceptable proof to just solve the equation which would prove/disprove the ...
818 views

### The sum of square roots of non-perfect squares is never integer [duplicate]

My question looks quite obvious, but I'm looking for a strict proof for this: Why can't the sum of two square roots of non-perfect squares be an integer? For example: $\sqrt8+\sqrt{15}$ isn't an ...
266 views

### Prove that $\sqrt{7}+\sqrt{3}$ is irrational [duplicate]

Is there a method by which we can prove that $$\sqrt{3}+\sqrt{7}$$ is irrational. It's obviously an irrational number, but I want to prove that mathematically.
1k views

### Proving that $\sqrt{3} +\sqrt{7}$ is rational/irrational [duplicate]

I took $\sqrt{3}+\sqrt{7}$ and squared it. This resulted in a new value of $10+2\sqrt{21}$. Now, we can say that $10$ is rational because we can divide it with $1$ and as for $2\sqrt{21}$, we divide ...
271 views

### Is $\sqrt{2}+\sqrt{3}$ rational or irrational? [duplicate]

I have proven that the sum of any two irrational numbers is not always irrational and that a rational plus an irrational is irrational, however I am not sure how to prove this specific case. I was ...
221 views

### proving $\sqrt 2 + \sqrt 3$ is irrational [duplicate]

I need to proof that $\sqrt{3} + \sqrt{2}$ is irrational, without using the fact that an irrational number plus a rational number equals irrational. also, i can't use the rational root theorem. that's ...
### $\sqrt{m}+\sqrt{n}\notin\mathbb{Q}$ Irrationals, Roots Proof [duplicate]
Let $n,m\in\mathbb{N}_0$ and $\sqrt{n}\not\in\mathbb{Q}$. Conclude that $\sqrt{n}+\sqrt{m}\not\in\mathbb{Q}$ What would you recommend me to read to be able to solve this? Because apart from the case ...