If p and q are real number such that p + q = 3, p ^ 4 + q ^ 4 = 369...

Given,

${p}^{4} + {q}^{4} = 369 ⇒ {\left({p}^{2} + {q}^{2}\right)}^{2} - 2{\left(pq\right)}^{2} = 369$

$⇒ {\left({\left(p + q\right)}^{2} - 2pq\right)}^{2} - 2{\left(pq\right)}^{2} = 369$

$⇒ {\left(9 - 2pq\right)}^{2} - 2{\left(pq\right)}^{2} = 369$

$⇒ {\left(pq\right)}^{2} - 18\left(pq\right) - 144 = 0$ $⇒ pq = 24 ( \text{rejected} ) , - 6$

Now, ${\left(\dfrac{1}{p} + \dfrac{1}{q}\right)}^{- 2} = \dfrac{{\left(pq\right)}^{2}}{{\left(p + q\right)}^{2}} = \dfrac{{\left(pq\right)}^{2}}{9} = 4$