Question

Se (R + 1/R)² = 3, então R³ + 1/R³ é:
a) 0
b) 1
c) 2
d) 3
e) 6

Answer 1

${(r+\dfrac{1}{r})}^{2}= 3 \\ {r}^{2} + 2 + \frac{1}{ {r}^{2} } = 3 \\ {r}^{2} + \frac{1}{ {r}^{2} } = 3 - 2$

${r}^{2} + \frac{1}{ {r}^{2} } = 1$

${r}^{3} + \frac{1}{ {r}^{3} } = (r + \frac{1}{r} )( {r}^{2} - 1 + \frac{1}{ {r}^{2} } )$

${(r + \frac{1}{r} )}^{2} = 3 \rightarrow \: r + \frac{1}{r} = \sqrt{3}$

${r}^{3} + \frac{1}{ {r}^{3} } = \sqrt{3} ( {r}^{2} + \frac{1}{ {r}^{2} } - 1) \\ \mathsf{ {r}^{3} + \frac{1}{ {r}^{3} } = \sqrt{3} .(1 - 1) = 0}$

$\texttt{Alternativa\,a}$