Question

2) Se x+1/x=3, então o valor de x^3+1/x^3 é:

Answer 1

Olá!

Desenvolvendo a expressão a ser determinada,

$\\ \mathsf{x^3 + \frac{1}{x^3} = \left ( x + \frac{1}{x} \right ) \cdot \left ( x^2 - x \cdot \frac{1}{x} + \frac{1}{x^2} \right )} \\\\\\ \mathsf{x^3 + \frac{1}{x^3} = \left ( x + \frac{1}{x} \right ) \cdot \left ( x^2 - 1 + \frac{1}{x^2} \right )} \\\\\\ \mathsf{x^3 + \frac{1}{x^3} = \left ( x + \frac{1}{x} \right ) \cdot \left ( x^2 + \frac{1}{x^2} - 1 \right )}$
 
 Elevando a equação, dada no enunciado, ao quadrado,

$\\ \mathsf{\left ( x + \frac{1}{x} \right )^2 = 3^2} \\\\\\ \mathsf{x^2 + 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 9} \\\\\\ \mathsf{x^2 + \frac{2x}{x} + \frac{1}{x^2} = 9} \\\\\\ \mathsf{x^2 + 2 + \frac{1}{x^2} = 9} \\\\\\ \boxed{\mathsf{x^2 + \frac{1}{x^2} = 7}}$
 
 Substituindo,

$\\ \mathsf{x^3 + \frac{1}{x^3} = \left ( x + \frac{1}{x} \right ) \cdot \left ( x^2 + \frac{1}{x^2} - 1 \right )} \\\\\\ \mathsf{x^3 + \frac{1}{x^3} = 3 \cdot \left(7-1\right)} \\\\\\ \mathsf{x^3 + \frac{1}{x^3} = 3 \cdot 6} \\\\\\ \boxed{\boxed{\mathsf{x^3 + \frac{1}{x^3} = 18}}}$
 
 
 A saber, $\mathsf{(a^3 + b^3) = (a + b) \cdot (a^2 - ab + b^2)}$.

Answer 2

x + 1/x = 3 => (x + 1/x)³ = 3³ => x³ + 3x².1/x + 3x(1/x)² + (1/x)³ = 3³
x³ + 1/x³ + 3x + 3/x + 1/x³ = 27
x³ + 1/x³ +3(x + 1/x) = 27
x³ + 1/x³ + 3. 3 = 27 
x³ + 1/x³ = 27 - 9
x³ +1/x³ = 18