# Linked Questions

9answers
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. ...
6answers
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 ...
6answers
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: $a^2/b^2 = \sqrt{40}$.
3answers
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 ...
3answers
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 ...
3answers
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.
3answers
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 ...
6answers
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 ...
3answers
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 ...
2answers
139 views

### $\sqrt{m}+\sqrt{n}\notin\mathbb{Q}$ 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 ...
0answers
146 views

Prove that the following number is irrational: $$\sqrt { 5 } +\sqrt { 3 }$$ Steps I took: Proof by contradiction: let us assume that $\sqrt { 5 } +\sqrt { 3 }$ is rational. If $\sqrt { 5 } +\... 1answer 80 views ### Both 2^(0.5) is irrational. [duplicate] Isn't 2^(0.5) rational? Method for proving: Contradiction. So show not P:2^(0.5) is rational. 0answers 70 views ### Sum of radicals is rational only for perfect squares [duplicate] How can we prove that if$a_1\sqrt{n_1}+a_2\sqrt{n_2}+\dots+a_k\sqrt{n_k}\in\mathbb{Q}$with$a_1,a_2\dots, a_k\in\mathbb{Q}^{*}_{+}$, then the natural numbers$n_1,n_2,\dots, n_k$are perfect squares?... 5answers 5k views ### Can a finite sum of square roots be an integer? Can a sum of a finite number of square roots of integers be an integer? If yes can a sum of two square roots of integers be an integer? The square roots need to be irrational. 1answer 3k views ### Is$\sqrt1+\sqrt2+\dots+\sqrt n$ever an integer? Related: Can a sum of square roots be an integer? Except for the obvious cases$n=0,1$, are there any values of$n$such that$\sum_{k=1}^n\sqrt k\$ is an integer? How does one even approach such a ...

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