Prove that 2√3 + √5 is an irrational number. Also check whether
(2√3 + √5). (2√3 - √5) is rational or irrational.
We have to prove that is irrational.
We can prove the above result by the method of contradiction as-
Let be a rational number which can be expressed in the form of p/q where q is not equal to 0 and p and q are relatively prime.
Hence,
Since p and q are integers and hence is a rational number.
So this implies that is a rational number which contradicts the fact that is irrational.
Hence our assumption is wrong and is irrational.
Now,
Hence this is a rational number.