If-x-1-x-3-x-4-1-x-4-

Question Number 90406 by I want to learn more last updated on 23/Apr/20

If x - \frac{1}{x} = 3

x^{4} - \frac{1}{x^{4}} = ? ? ?

Answered by som(math1967) last updated on 23/Apr/20

x - \frac{1}{x} = 3

\Rightarrow {(x - \frac{1}{x})}^{2} = 9

\Rightarrow {(x + \frac{1}{x})}^{2} - 4 x . \frac{1}{x} = 9

\Rightarrow (x + \frac{1}{x}) = \pm \sqrt{13}

a g a i n {(x - \frac{1}{x})}^{2} = 9

∴ x^{2} + \frac{1}{x^{2}} = 9 + 2 = 11

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

= \pm 33 \sqrt{13} a n s

Commented by I want to learn more last updated on 23/Apr/20

I appreciate sir

Answered by MJS last updated on 23/Apr/20

x = u + v = \frac{{(u + v)}^{2} (u - v)}{(u + v) (u - v)}

\frac{1}{x} = \frac{u - v}{(u + v) (u - v)}

x^{4} - \frac{1}{x^{4}} = \frac{{(u + v)}^{8} {(u - v)}^{4} - {(u - v)}^{4}}{{(u + v)}^{4} {(u - v)}^{4}} =

= \frac{{(u^{2} - v^{2})}^{4} {(u + v)}^{4} - {(u - v)}^{4}}{{(u^{2} - v^{2})}^{4}}

x - \frac{1}{x} = 3 \Rightarrow x^{2} - 3 x - 1 = 0 \Rightarrow x = \frac{3}{2} \pm \frac{\sqrt{13}}{2}

u = \frac{3}{2} \land v^{2} = \frac{13}{4}

\frac{{(\frac{9}{4} - \frac{13}{4})}^{4} {(u + v)}^{4} - {(u - v)}^{4}}{{(\frac{9}{4} - \frac{13}{4})}^{4}} =

= {(u + v)}^{4} - {(u - v)}^{4} = 8 u v (u^{2} + v^{2}) =

= 8 u v (\frac{9}{4} + \frac{13}{4}) = 44 u v = \pm 33 \sqrt{13}

Commented by I want to learn more last updated on 23/Apr/20

I appreciate sir