In a test I tried to solve recently I came across the following question:
Prove $$\sqrt3 + \sqrt[3]{2}$$ is irrational
I tried proving it by saying it is equal to some rational number $$\sqrt3 + \sqrt[3]{2} = \frac{m}{n}$$ and reaching a contradiction.
I tried squaring / cubing both sides and that didnt work. So how do I go about solving this problem?
Best Answer
It might be useful to first prove the following two statements - the sum of a rational and irrational number is irrational, and the product of a rational and irrational number is irrational. You can then move the $\sqrt 3$ and get:
$\sqrt[3] 2=\frac{m}{n}-\sqrt 3$
$2=(\frac{m}{n}-\sqrt 3)^3$
After expanding the term at the RHS, the two lemmas you proved might come in handy.