Question

If x+1 x = 5 , then evaluate x⁴ +1 x⁴

Answer 1

$$\begin{gathered} x+\frac{1}{x}=5 \\ \Rightarrow {\left(x+\frac{1}{x}\right)}^2=5^2\{\mathrm{squaring}\mathrm{both}\mathrm{sides}\} \\ \Rightarrow x^2+\frac{1}{x^2}+2\times x\times \frac{1}{x}=25 \\ \Rightarrow x^2+\frac{1}{x^2}+2=25 \\ \Rightarrow x^2+\frac{1}{x^2}=23 \\ \Rightarrow {\left(x^2+\frac{1}{x^2}\right)}^2={\left(23\right)}^2\{\mathrm{again}\mathrm{squaring}\mathrm{both}\mathrm{sides}\} \\ \Rightarrow x^4+\frac{1}{x^4}+2\times x^2\times \frac{1}{x^2}=529 \\ \Rightarrow x^4+\frac{1}{x^4}+2=529 \\ \Rightarrow x^4+\frac{1}{x^4}=527 \end{gathered}$$