Question

If $x^4+\frac{1}{x^4}=194$, find $x^3+\frac{1}{x^3}$, $x^2+\frac{1}{x^2}$ and $x+\frac{1}{x}$

Answer 1

Given,

$\Rightarrow x^4+\frac{1}{x^4}=194$

$\Rightarrow x^4+\frac{1}{x^4}+2=194+2$

$\Rightarrow {(x^2+\frac{1}{x^2})}^2=196$

$\therefore x^2+\frac{1}{x^2}=14$.......$\left(ii\right)$

$\Rightarrow x^2+\frac{1}{x^2}+2=14+2$

${(x+\frac{1}{x})}^2=16$

$\therefore x+\frac{1}{x}=4$.......$\left(iii\right)$

$\Rightarrow {(x+\frac{1}{x})}^3=x^3+\frac{1}{x^3}+3(x+\frac{1}{x})$

$\Rightarrow 4^3=x^3+\frac{1}{x^3}+3\left(4\right)$

$\therefore x^3+\frac{1}{x^3}=52$........$\left(i\right)$