if-x-4-1-x-4-322-find-x-3-1-x-3-

Question Number 26228 by ktomboy1992 last updated on 23/Dec/17

if x^{4} + \frac{1}{x^{4}} = 322 find x^{3} - \frac{1}{x^{3}}

Commented by kaivan.ahmadi last updated on 22/Dec/17

x^{4} + \frac{1}{x^{4}} = {(x^{2} + \frac{1}{x^{2}})}^{2} - 2 \Rightarrow

x^{2} + \frac{1}{x^{2}} = \sqrt{324} = 18 = {(x + \frac{1}{x})}^{2} - 2 \Rightarrow

x + \frac{1}{x} = \sqrt{20}

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

20 \sqrt{20} - 3 \sqrt{20} = 17 \sqrt{20}