Question

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

Answer 1

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

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

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

So, 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♧♧

Answer 2

♧♧HERE IS YOUR ANSWER♧♧

Given :
x = 2 + \sqrt{3}x=2+√​3​​​ 

Now,
\begin{lgathered}\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}\end{lgathered}​​x​​1​​=​2+√​3​​​​​1​​​=​(2+√​3​​​)(2−√​3​​​)​​2−√​3​​​​​​=​4−3​​2−√​3​​​​​​=2−√​3​​​​​ 

Again,
\begin{lgathered}x + \frac{1}{x} \\ = (2 + \sqrt{3} ) + (2 - \sqrt{3} ) \\ = 4\end{lgathered}​x+​x​​1​​​=(2+√​3​​​)+(2−√​3​​​)​=4​​ 

So, L.H.S.
\begin{lgathered}= {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\end{lgathered}​=x​3​​+​x​3​​​​1​​​=(x+​x​​1​​)((x+​x​​1​​)​2​​−3)​=4(4​2​​−3)​=4(16−3)​=4×13​=52​​ 
= R.H.S. [Proved]

♧♧HOPE THIS HELPS YOU♧♧