Question

Considere o valor da expressão $x- \frac{1}{x} = \frac{9}{5}$ . Determine:

a)$x^{4} + \frac{1}{x^4} =$

Answer 1

Olá.

$x-\dfrac{1}{x}=\dfrac{9}{5}\\\\\\\mathsf{x-\dfrac{1}{x}=\dfrac{9}{5}}$

$x-x^{-1}=\dfrac{9}{5}\\\\(x-x^{-1})^2=(\dfrac{9}{5})^2\\\\(x^2-2\cdot x\cdot x^{-1}-x^{-2})=\dfrac{81}{25}\\\\x^2-2\cdot\dfrac{x}{x}-x^{-2}=\dfrac{81}{25}\\\\x^2-2-x^{-2}=\dfrac{81}{25}\\\\\boxed{x^2-x^{-2}=\dfrac{81}{25}+2}\ \ \ \mathbf{ou...\ \ \ }\boxed{x^2-\dfrac{1}{x^2}=\dfrac{81}{25}+2}$


$\\x^4 + \dfrac{1}{x^4}=( x^4 + 2 + \dfrac{1}{x^4} \right ) - 2 = ( x^2 + \dfrac{1}{x^2} )^2-2$

$( x^2 + \dfrac{1}{x^2})^2 - 2 = ( \dfrac{131}{25})^2 - 2 =\dfrac{17161}{625} - 2 =\dfrac{17161 - 1250}{625} =\boxed{\dfrac{15911}{625}}$

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