Question

If x=(2+root 3), show that (x3 +1/x3)=52

Answer 1

♧♧HERE IS YOUR ANSWER♧♧

Given :
$x = 2 + \sqrt{3}$

Now,
$\frac{1}{x} = \frac{1}{2 + \sqrt{3} } \\ = \frac{2 - \sqrt{3} }{(2 + \sqrt{3} )( 2 - \sqrt{3} )} \\ = \frac{2 - \sqrt{3} }{4 - 3} \\ = 2 - \sqrt{3}$

So,
$x + \frac{1}{x } = (2 + 3 \sqrt{3} ) + (2 - \sqrt{3} ) \\ = 4$

Now, L.H.S.
$= {x}^{3} + \frac{1}{ {x}^{3} } \\ = (x + \frac{1}{x} )( {(x + \frac{1}{x}) }^{2} - 3) \\ = 4( {4}^{2} - 3) \\ = 4(16 - 3) \\ = 4 \times 13 \\ = 52$
=R.H.S. [Proved]

♧♧HOPE THIS HELPS YOU♧♧