if x+ 1 ÷x=6, find the value of x4 + 1 / x⁴

Answer :

We have  : $x + \frac{1}{x} = 6$

Taking whole square on both hand side we get

$$\begin{gathered} {\left(x + \frac{1}{x}\right)}^2 = 6^2 \\ \Rightarrow x^2 + \frac{1}{x^2} + 2 x \times \frac{1}{x} = 36 \\ \Rightarrow x^2 + \frac{1}{x^2} + 2 = 36 \\ \Rightarrow x^2 + \frac{1}{x^2} = 34 \\ \mathrm{Taking} \mathrm{whole} \mathrm{square} \mathrm{on} \mathrm{both} \mathrm{hand} \mathrm{side} \mathrm{we} \mathrm{get} \\ \Rightarrow {\left({\mathrm{x}}^2 + \frac{1}{{\mathrm{x}}^2}\right)}^2 = {34}^2 \\ \Rightarrow {\mathrm{x}}^4 + \frac{1}{{\mathrm{x}}^4} + 2{\mathrm{x}}^2 \times \frac{1}{{\mathrm{x}}^2} = 1156 \\ \Rightarrow {\mathrm{x}}^4 + \frac{1}{{\mathrm{x}}^4} + 2 = 1156 \\ \mathbf{\Rightarrow }{\mathbf{x}}^{\mathbf{4}}\mathbf{ }\mathbf{+}\mathbf{ }\frac{\mathbf{1}}{{\mathbf{x}}^{\mathbf{4}}}\mathbf{ }\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{1154}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{(}\mathbf{ }\mathbf{Ans}\mathbf{ }\mathbf{)} \end{gathered}$$