Mathematics

The key insight is that $p$ and $q$ are reciprocals of each other—notice how the numerator and denominator swap between them. Rather than rationalizing each fraction separately, find $p + q$ first by combining them over a common denominator, then use the algebraic identity $(p+q)^2 = p^2 + 2pq + q^2$ to isolate $p^2 + q^2$. This avoids messy radical arithmetic and lets the structure of the problem do the work.

The key insight is that $p$ and $q$ are reciprocals of each other—notice how the numerator and denominator swap between them. Rather than rationalizing each fraction separately, find $p + q$ first by combining them over a common denominator, then use the algebraic identity $(p+q)^2 = p^2 + 2pq + q^2$ to isolate $p^2 + q^2$. This avoids messy radical arithmetic and lets the structure of the problem do the work.