If-a-x-9-x-9-a-7-find-a-x-9-1-4-x-9-a-1-4-

Question Number 182530 by HeferH last updated on 10/Dec/22

I f : \frac{a}{x^{9}} + \frac{x^{9}}{a} = 7

f i n d : \sqrt[4]{\frac{a}{x^{9}}} + \sqrt[4]{\frac{x^{9}}{a}}

Answered by mr W last updated on 10/Dec/22

l e t t = \frac{a}{x^{9}}

s a y \sqrt[4]{t} + \sqrt[4]{\frac{1}{t}} = x > 0

\Rightarrow \sqrt{t} + \sqrt{\frac{1}{t}} = x^{2} - 2

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

\Rightarrow 7 = {(x^{2} - 2)}^{2} - 2

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

\Rightarrow x^{2} - 2 = 3 (- 3 r e j e c t e d)

\Rightarrow x^{2} = 5

\Rightarrow x = \sqrt{5} (- \sqrt{5} r e j e c t e d)

i . e . \sqrt[4]{\frac{a}{x^{9}}} + \sqrt[4]{\frac{x^{9}}{a}} = \sqrt{5}

Answered by Frix last updated on 10/Dec/22

t + \frac{1}{t} = 7 \Rightarrow t = φ^{4} \Rightarrow answer is \sqrt{5}

Commented by manxsol last updated on 11/Dec/22

S i r F r i x w h a t i s φ^{4}, w h a t i s t h e t h e o r y ?

Commented by Frix last updated on 11/Dec/22

φ = \frac{1 + \sqrt{5}}{2}; φ^{4} = \frac{7 + 3 \sqrt{5}}{2}

\frac{1}{φ} = \frac{- 1 + \sqrt{5}}{2}; \frac{1}{φ^{4}} = \frac{7 - 3 \sqrt{5}}{2}

Commented by manxsol last updated on 12/Dec/22

t h a n k s, S i r F r i x i s g o l d e n n u m b e r . c h e c k m y n e x t q u e s t i o n

Answered by Ar Brandon last updated on 11/Dec/22

\frac{a}{x^{9}} + \frac{x^{9}}{a} = 7 \Rightarrow {(\sqrt{\frac{a}{x^{9}}} + \sqrt{\frac{x^{9}}{a}})}^{2} - 2 = 7

\Rightarrow \sqrt{\frac{a}{x^{9}}} + \sqrt{\frac{x^{9}}{a}} = 3

\Rightarrow {(\sqrt[4]{\frac{a}{x^{9}}} + \sqrt[4]{\frac{x^{9}}{a}})}^{2} - 2 = 3

\Rightarrow \sqrt[4]{\frac{a}{x^{9}}} + \sqrt[4]{\frac{x^{9}}{a}} = \sqrt{5}