If (p + q)^-1(p^-1 + q^-1) = p^aq^b, prove that a + b +2 = 0, where p and q are different positive primes.

If $\Big(\dfrac{p^{-1}q^{2}}{p^2q^{-4}}\Big)^7 ÷ \Big(\dfrac{p^3q^{-5}}{p^{-2}q^{3}}\Big)^{-5} = p^xq^y$, find x + y, where p and q are different positive primes.

Solve the following equation for x:

$(13)^{\sqrt{x}} = 4^4 - 3^4 - 6$

Solve the following equation for x:

$\Big(\sqrt{\dfrac{3}{5}}\Big)^{x + 1} = \dfrac{125}{27}$