Question

Sendo $(x + \frac{1}{x} )^{2} = 3$ , Calcule $x^{3} + \frac{1}{x^{3}}$ Por favor é um trabalho

Answer 1

$(x+ \frac{1}{x} )^2=3 \\ \\ (x+ \frac{1}{x} )= \sqrt{3} \\ \\ elevando~~ambos~~termos~~ ao~~cubo \\ \\ (x+ \frac{1}{x} )^3= \sqrt{3^3} \\ \\ calculando~cubo~~da~~soma \\ \\ x^3+3.x^2 \frac{1}{x} +3.x. \frac{1}{x^2} + \frac{1}{x^3} =3 \sqrt{3} \\ \\ x^3+3x^{\not 2}. \frac{1}{\not x} +3.\not x. \frac{1}{x^{\not2}} + \frac{1}{x^3} =3 \sqrt{3} \\ \\ \\ x^3+ \frac{1}{x^3} +3(x+ \frac{1}{x})=3 \sqrt{3} \\ \\ substituir~~(x+ \frac{1}{x})~~por~~ \sqrt{~3}$

$x^3+ \frac{1}{x^3} +3 \sqrt{3} =3 \sqrt{3} \\ \\ x^3+ \frac{1}{x^3} =3 \sqrt{3} -3 \sqrt{3} \\ \\ \fbox{ x^3+ \frac{1}{x^3} =0 }$