Given that p=1+√2 p = 1 + 2 and q=1−√2, q = 1 − 2 , evaluate (p2−q2)2pq ( p 2 − q 2 ) 2 p q .

Given that $p=1+\sqrt{2}$ and $q=1-\sqrt{2},$ evaluate $\frac{(p^2-q^2)}{2pq}$.

Answer Details

First, let's evaluate p^2 and q^2: p^2 = (1+√2)^2 = 1 + 2√2 + 2 = 3 + 2√2 q^2 = (1-√2)^2 = 1 - 2√2 + 2 = 3 - 2√2 Next, let's substitute the values of p^2 and q^2 into the expression (p^2 - q^2)/(2pq): (p^2 - q^2)/(2pq) = ((3 + 2√2) - (3 - 2√2))/(2(1+√2)(1-√2)) = (4√2)/(2(1-2)) = (4√2)/(-2) = -2√2 Therefore, the answer is -2√2.