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 For finding the value of  x^{6}+ \frac{1}{x^6}
we can take
x+ \frac{1}{x}=5
(x+ \frac{1}{x})^3=(5)^3 

x^3+ \frac{1}{x^3}+3*x* \frac{1}{x}(x+ \frac{1}{x})=125
x^3+ \frac{1}{x^3}+3*5=125
x^3+ \frac{1}{x^3} =125-15
x^3+ \frac{1}{x^3} =110
(x^3+ \frac{1}{x^3})^2=(110)^2
(x^3)^2+2*x^3* \frac{1}{x^3}+( \frac{1}{x^3})^2= 12100
x^6+ \frac{1}{x^6}=12100-2
x^6+ \frac{1}{x^6} =12098

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