If x² - √7x + 1 = 0, then what is the value of $${x^5} + \frac{1}{{{x^5}}}?$$

$$\eqalign{ & {x^2} - \sqrt 7 x + 1 = 0 \cr & {x^2} + 1 = \sqrt 7 x \cr & x + \frac{1}{x} = \sqrt 7 \cr & {x^5} + \frac{1}{{{x^5}}} = \left( {{x^2} + \frac{1}{{{x^2}}}} \right)\left( {{x^3} + \frac{1}{{{x^3}}}} \right) - \left( {x + \frac{1}{x}} \right) \cr & = \left( {{{\left( {\sqrt 7 } \right)}^2} - 2} \right)\left( {{{\left( {\sqrt 7 } \right)}^3} - 3 \times \sqrt 7 } \right) - \sqrt 7 \cr & = 5 \times 4\sqrt 7 - \sqrt 7 \cr & = 19\sqrt 7 \cr} $$