Question

if x²+1 = 5/2. find x³ + 1/x³

Answer 1

${x}^{2} + 1 = \frac{5}{2}$
${x}^{2} = \frac{5}{2} - 1 = \frac{3}{2} = > x = \sqrt{ \frac{3}{2} }$
${x}^{3} + \frac{1}{ {x}^{3} } = {( \sqrt{ \frac{3}{2} } )}^{3} + \frac{1}{ {( \sqrt{ \frac{3}{2} } )}^{3} }$
$= \sqrt{ \frac{ {3}^{3} }{{2}^{3}}} + \sqrt{ \frac{ {2}^{3} }{{3}^{3}}}$
$= \sqrt{ \frac{ 27 }{8}} + \sqrt{ \frac{ 8}{27}}$
$= \frac{ \sqrt{ {27}^{2} } + \sqrt{ {8}^{2} } }{ \sqrt{ {8 \times 27} }}= \frac{ \sqrt{ {27}^{2} } + \sqrt{ {8}^{2} } }{ \sqrt{ {2}^{2} \times 2 \times {3}^{2} \times 3} }$
$\frac{27 + 8}{6 \sqrt{6} } = \frac{35}{6 \sqrt{6} }$