Question

if X=(√5+√3)/(√5-√3)=1/y then the value of x^3+y^3=?

Answer 1

$x = \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } = \frac{1}{y}$
Now let's rationalise the terms,

$x = \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } \times \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} + \sqrt{3} } \\ = \frac{ {( \sqrt{5} + \sqrt{3} ) }^{2} }{ {{ \sqrt{5} }^{2} - { \sqrt{3} }^{2} } } \\ = \frac{8 + 2 \sqrt{15} }{2} \\ = \frac{2(4 + \sqrt{15}) }{2} \\ = 4 + \sqrt{15}$


Now,
here,
$x = 4 + \sqrt{15} = \frac{1}{y} \\ y = \frac{1}{4 + \sqrt{15} }$


Again, rationalise the term
$y = \frac{1}{ 4 + \sqrt{15} } \times \frac{4 - \sqrt{15} }{4 - \sqrt{15} } \\ = \frac{4 - \sqrt{15} }{16 - 15} \\ = 4 - \sqrt{15}$

Therefore,

$x = 4 + \sqrt{15} \\ \\ and \\ \\ y = 4 - \sqrt{15}$


${x}^{3} + {y}^{3} = {(x + y)}^{3} - 3xy(x + y)$



Substitute for x and y and you'll get,

${x}^{3} + {y}^{3} \\ = (4 + \sqrt{15} +4 - \sqrt{15} ) - 3(4 + \sqrt{15} )(4 - \sqrt{15} )(4 + \sqrt{15} + 4 - \sqrt{15} ) \\ = 8 - 3(16 - 15)(8) \\ = 8 - 24 \\ = - 16$


Therefore your answer will be -16.