Question

Se
$x + \frac{1}{x} = 5$
Quanto é
${x}^{3} + \frac{1}{ {x}^{3} }$

Answer 1

Vamos elevar os dois lados ao cubo.

$\left( x + \dfrac{1}{x} \right)^3=5^3 \\ \\ \left( x + \dfrac{1}{x} \right)^2 \times \left(x+\dfrac{1}{x}\right)=5^3 \\ \\ \left( x^2 +2 + \dfrac{1}{x^2} \right) \times \left(x+\dfrac{1}{x}\right)=5^3 \\ \\ \left( x^3+x+2x+\dfrac{2}{x}+\dfrac{x}{x^2}+\dfrac{1}{x^3}\right)=5^3 \\ \\ \left( x^3+3x+\dfrac{2}{x}+\dfrac{1}{x}+\dfrac{1}{x^3}\right)=5^3 \\ \\ \left( x^3+3x+\dfrac{3}{x}}+\dfrac{1}{x^3}\right)=5^3 \\ \\ \left( x^3+\dfrac{1}{x^3}+3x+\dfrac{3}{x}}\right)=5^3$
$\left( x^3+\dfrac{1}{x^3}+3 \left(x +\dfrac{1}{x}} \right)\right)=5^3 \\ \\ \left( x^3+\dfrac{1}{x^3}+3 \times 5\right)=5^3 \\ \\ x^3+\dfrac{1}{x^3}+15=5^3 \\ \\ x^3+\dfrac{1}{x^3}=5^3 - 15\\ \\ \boxed{x^3+\dfrac{1}{x^3}=110}$