Question

if x=2+√3, find the value of x cube + 1 /x cube​

Answer 1

$\huge\sf{solution:}$

★$x = 2 + \sqrt{3}$

and

★$\frac{1}{x} = 2 - \sqrt{3}$

then find the value of :-

${x}^{3} + \frac{1}{ {x}^{3} }$

Using a suitable identity :-

★ $\boxed{\sf{ </p><p>(x + \frac{1}{ {x} } )^{3} = {x}^{3} + \frac{1}{ {x}^{3} } + 3x \times \frac{1}{x} (x + \frac{1}{x})}}$

put the value of x and $\frac{1}{x}$

$\implies$ $(2 + \sqrt{3} + 2 - \sqrt{3} ) ^{3} = {x}^{3} + \frac{1}{ {x}^{3} } + 3.1.(2 + \sqrt{3} + 2 - \sqrt{3} )$

$\implies$ $(4) ^{3} = {x}^{3} + \frac{1}{ {x}^{3} } + 3 \times 4$

$\implies$ $64 - 12 = {x}^{3} + \frac{1}{ {x}^{3} }$

$\implies$ ${x}^{3} + \frac{1}{ {x}^{3} } = 52$

the required answer is 52.