Prove by contradiction

Need help with this question. Thanks

What have you tried? The proof is pretty much the same as for $\sqrt{2}$.

I cubed both sides and ended up with 2=a^3/b^3. I then put 2a^3=b^3. I then said that b=(2n)^3

You are jumping across landscapes here with this argument. Too many missing links to a point where you're confusing yourself.

Firstly, you need to hammer down the fact that you assume cube root 2 is rational, this means expressing it in the form a/b where a and b are irreducible. And that's a key word.

Secondly, you cube both sides and get 2 = a^3/b^3 and this is the same as 2b^3 = a^3. You got this the wrong way round in yours. Be careful!
So, we know that a^3 is even. What does this tell us about a itself?

A is even.

a=2n^3

Nope. Where did that come from?

i assumed since a was even and the equation has a^3, we would cube 2n

so 8n^3

Yes we do, but you cubed it incorrectly, and didn't write a^3 to indicate it. If you're cubing 2n you need to cube the 2 as well.

$(2n)^3 = 2^3n^3$

is it 8a^3=b^3

Nope.

We have $2b^3 = a^3$ so it becomes $2b^3 = 8n^3$.

this is the step I am unsure about. I would divide both sides by 2

I would say b is even.

Indeed. Do you see the contradiction now? Since a and b must both be even.

Have a look at my post #4 on this thread and what assumptions we have made if you're not seeing the contradiction immediately.

Indeed. Do you see the contradiction now? Since a and b must both be even.

Have a look at my post #4 on this thread and what assumptions we have made if you're not seeing the contradiction immediately.