If $$x + \frac{1}{x} = 5,$$   then what is the value of $${x^6} + \frac{1}{{{x^6}}}?$$

$$\eqalign{ & x + \frac{1}{x} = 5 \cr & {\text{Cubing both sides, we get}} \cr & {x^3} + \frac{1}{{{x^3}}} + 3 \times x \times \frac{1}{x}\left( {x + \frac{1}{x}} \right) = 125 \cr & {x^3} + \frac{1}{{{x^3}}} + 3 \times 5 = 125 \cr & {x^3} + \frac{1}{{{x^3}}} = 110 \cr & {\text{Squaring both sides, we get}} \cr & {x^6} + \frac{1}{{{x^6}}} + 2 = 12100 \cr & {x^6} + \frac{1}{{{x^6}}} = 12098 \cr} $$