Assuming the truth of the theorem that stales that is irrational whenever n is a positive integer that is not a perfect square, prove that is irrational.

Solution:Step 1In this problem we have to prove that is irrational assuming the truth of the theorem that states that is irrational whenever n is positive integer and is not a perfect square.We will prove this problem by method of contradiction.Any number which can be written in the form of where are integer and .Step 2Let us assume that is rational.We have where p and q are integer and q is not equal to 0.On squaring both side of equation we get Or, +3=. [using formula Or, On subtracting 5 on both side of equation we get On dividing both side by 2 we get