If x = 4 + \(\sqrt {7}\), y = 4 - \(\sqrt {7}\) and z = 1, then find the value of \(\frac{x}{yz}\) + \(\frac{y}{xz}\) + \(\frac{z}{xy}\) + 2[\(\frac{1}{x}\) + \(\frac{1}{y}\) + \(\frac{1}{z}\)].

9

x = 4 + \(\sqrt {7}\), y = 4 - \(\sqrt {7}\) and z = 1

⇒ \(\frac{x}{yz}\) + \(\frac{y}{xz}\) + \(\frac{z}{xy}\) + 2[\(\frac{1}{x}\) + \(\frac{1}{y}\) + \(\frac{1}{z}\)]

= \(\frac{x^2 + y^2 + z^2 + 2xy + 2yz + 2zx}{xyz}\)

= \(\frac{(x + y + z)^2}{xyz}\)

= \(\frac{(4 + \sqrt {7} + 4 - \sqrt {7} + 1)^2}{(4 + \sqrt {7}) (4 - \sqrt {7})( -1)}\)

= \(\frac{81}{16 - 7}\)

= \(\frac{81}{9}\) = 9