If: (a/x^9 ) + (x^9 /a) = 7 find: ((a/x^9 ))^(1/4) + ((x^9 /a))^(1/4)


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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

	$\frac{a}{x^{9}} + \frac{x^{9}}{a} = 7 \Rightarrow {(\sqrt{\frac{a}{x^{9}}} + \sqrt{\frac{x^{9}}{a}})}^{2} - 2 = 7$ $\Rightarrow \sqrt{\frac{a}{x^{9}}} + \sqrt{\frac{x^{9}}{a}} = 3$ $\Rightarrow {(\sqrt[4]{\frac{a}{x^{9}}} + \sqrt[4]{\frac{x^{9}}{a}})}^{2} - 2 = 3$ $\Rightarrow \sqrt[4]{\frac{a}{x^{9}}} + \sqrt[4]{\frac{x^{9}}{a}} = \sqrt{5}$
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