please-help-me-with-this-confusing-question-x-2x-y-y-y-x-4-1-xy-xy-yx-16-2-solve-for-x-and-y-

Question Number 17638 by chux last updated on 08/Jul/17

please help me with this

confusing question

x^{2 x / y} \times y^{y / x} = 4 \dots \dots (1)

{(xy)}^{xy + yx} = 16 \dots . . (2)

solve for x and y

Commented by mrW1 last updated on 09/Jul/17

x = 1 and y = 2

Commented by chux last updated on 09/Jul/17

please can it be solved analytically .

Commented by mrW1 last updated on 09/Jul/17

from (2) you get

{[{(xy)}^{(xy)}]}^{2} = 16

{(xy)}^{(xy)} = 4

(xy) = 2

from (1) you get

x^{2 x / y} = {4 y}^{- y / x}

{(x^{2 x / y})}^{xy} = {({4 y}^{- y / x})}^{xy}

x^{{2 x}^{2}} = {16 y}^{- y^{2}}

x^{{2 x}^{2}} = 16 {(\frac{x}{2})}^{\frac{2}{x^{2}}}

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

\Rightarrow x = 1 and 1 - \frac{1}{x^{2}} = 0

Commented by chux last updated on 09/Jul/17

wow!! thank you so much .

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