Question

Prove that
$\: prove \: \: that \: \: \sqrt{2 + \sqrt{3} } \: \: is \: irrational$
​

Answer 1

Answer:

Step-by-step explanation:

Suppose \sqrt{2} + \sqrt{3} is rational, say r so that

\sqrt{2} + \sqrt{3} = r.

Squaring both sides, we have 2 + 2 \sqrt{6} + 3 = r^2 which means that \sqrt{6} = r^2 - 5.

Since the set of rational numbers is closed under multiplication and addition, r^2 - 5 is therefore rational. However, as we have proved in the previous post, \sqrt{6} is irrational. A contradiction!

Therefore, \sqrt{2} + \sqrt{3} is irrational.