# Prove the following statement by proving its contrapositive: if $r$ is irrational, then ${ r }^{ \frac { 1 }{ 5 } }$ is irrational

Just a disclaimer before I proceed with my question and the proof I wrote up: I know that this question has been asked before, for example here, but I am more interested in being critiqued on how I wrote the proof and its completeness. In addition, I am having trouble seeing how it proves what I first set out to prove.

Theorem: if $r$ is irrational, then ${ r }^{ \frac { 1 }{ 5 } }$ is irrational

Proof: We prove the contrapositive: if ${ r }^{ \frac { 1 }{ 5 } }$ is rational, then $r$ is rational

1) Assume that ${ r }^{ \frac { 1 }{ 5 } }$ is rational, then there exists $a,b\in\mathbb{Z}$ such that:

${ r }^{ \frac { 1 }{ 5 } }=\frac { a }{ b }$ where $a,b$ are coprime and $b\neq 0$

2) Therefore, ${ r }=\frac { a^{ 5 } }{ b^{ 5 } }$

3) If $a,b\in\mathbb{Z}$, then $a^5,b^5\in\mathbb{Z}$ as well.

4) Therefore $r\in\mathbb{Q}$

Q.E.D.

When it comes to proving that the square root of $2$ is irrational, I can quickly see and understand why $\sqrt { 2 }$ is indeed irrational. However, with this proof, I don't understand how this actually proves the theorem I set out to prove. Does proving it by using the contrapositive just make it that much simpler? Or did I miss some vital steps in my proof?

Please feel free to give me constructive criticism about my proof writing techniques as well.

• Your proof is great. One small thing is that you may want to note that $b^5 \neq 0$ since $b \neq 0$. – Christian Gaetz Apr 28 '16 at 12:51
• I agree with the comment above. $[p \implies q] \iff [\neg q \implies \neg p]$. – barak manos Apr 28 '16 at 12:53
• @user75296 So just edit step 3) to look like this? 3) if $a,b\in\mathbb{Z}$, then $a^5,b^5\in\mathbb{Z}$ where $b^5\neq 0$ as well. – Cherry_Developer Apr 28 '16 at 13:02

In this particular problem, the statement involves assuming $r$ is irrational. The problem is that we can't really say much about what $r$ looks like.
On the subject of your exposition, it looks great! The comment about noting that $b^5 \neq 0$ is really the only thing to improve on.