Question

if x = 4 - √15 find the value of x³ + 1/x³​

Answer 1

$\rm \huge\color{PALEVIOLETRED} Solution$

$\sf{x = 4 - \sqrt{15} }$

$\sf{ \frac{1}{x} = \frac{1}{ 4 - \sqrt{15} }}$

rationalise the denominator

$\sf{ \frac{1}{x} = \frac{1}{ 4 - \sqrt{15} } \times \frac{4 + \sqrt{15} }{4 + \sqrt{15} } }$

$\sf{ \frac{1}{x} = \frac{4 + \sqrt{15} }{ {4}^{2} - {(\sqrt{15} )}^{2} }}$

$\sf{ \frac{1}{x} = \frac{4 + \sqrt{15} }{ 16 - 15 }}$

$\sf{ \frac{1}{x} = 4 + \sqrt{15} }$

now,

$\sf{ x + \frac{1}{x} = 4 - \sqrt{15} + 4 + \sqrt{15} }$

$\sf{ x + \frac{1}{x} =8}$

cubing on both sides.

$\sf{ {(x + \frac{1}{x} )}^{3} ={8}^{3}}$

$\sf{{ {x}^{3} + \frac{1}{{x}^{3} } + 3(8 )= 512 }}$

$\sf{{ {x}^{3} + \frac{1}{{x}^{3} } = 512 - 24 }}$

$\fbox{ \sf{{ {x}^{3} + \frac{1}{{x}^{3} } = 488 }}}$